Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check...
Transcript of Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check...
![Page 1: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/1.jpg)
logo1
New Idea An Example Double Check
Laplace Transforms for Systems ofDifferential Equations
Bernd Schroder
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 2: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/2.jpg)
logo1
New Idea An Example Double Check
The Laplace Transform of a System
1. When you have several unknown functions x,y, etc., thenthere will be several unknown Laplace transforms.
2. Transform each equation separately.3. Solve the transformed system of algebraic equations for
X,Y, etc.4. Transform back.5. The example will be first order, but the idea works for any
order.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 3: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/3.jpg)
logo1
New Idea An Example Double Check
The Laplace Transform of a System1. When you have several unknown functions x,y, etc., then
there will be several unknown Laplace transforms.
2. Transform each equation separately.3. Solve the transformed system of algebraic equations for
X,Y, etc.4. Transform back.5. The example will be first order, but the idea works for any
order.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 4: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/4.jpg)
logo1
New Idea An Example Double Check
The Laplace Transform of a System1. When you have several unknown functions x,y, etc., then
there will be several unknown Laplace transforms.2. Transform each equation separately.
3. Solve the transformed system of algebraic equations forX,Y, etc.
4. Transform back.5. The example will be first order, but the idea works for any
order.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 5: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/5.jpg)
logo1
New Idea An Example Double Check
The Laplace Transform of a System1. When you have several unknown functions x,y, etc., then
there will be several unknown Laplace transforms.2. Transform each equation separately.3. Solve the transformed system of algebraic equations for
X,Y, etc.
4. Transform back.5. The example will be first order, but the idea works for any
order.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 6: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/6.jpg)
logo1
New Idea An Example Double Check
The Laplace Transform of a System1. When you have several unknown functions x,y, etc., then
there will be several unknown Laplace transforms.2. Transform each equation separately.3. Solve the transformed system of algebraic equations for
X,Y, etc.4. Transform back.
5. The example will be first order, but the idea works for anyorder.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 7: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/7.jpg)
logo1
New Idea An Example Double Check
The Laplace Transform of a System1. When you have several unknown functions x,y, etc., then
there will be several unknown Laplace transforms.2. Transform each equation separately.3. Solve the transformed system of algebraic equations for
X,Y, etc.4. Transform back.5. The example will be first order, but the idea works for any
order.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 8: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/8.jpg)
logo1
New Idea An Example Double Check
So Everything Remains As It Was
Time Domain (t)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 9: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/9.jpg)
logo1
New Idea An Example Double Check
So Everything Remains As It Was
Time Domain (t)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 10: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/10.jpg)
logo1
New Idea An Example Double Check
So Everything Remains As It Was
Time Domain (t)
OriginalDE & IVP
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 11: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/11.jpg)
logo1
New Idea An Example Double Check
So Everything Remains As It Was
Time Domain (t)
OriginalDE & IVP
-L
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 12: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/12.jpg)
logo1
New Idea An Example Double Check
So Everything Remains As It Was
Time Domain (t)
OriginalDE & IVP
Algebraic equation forthe Laplace transform
-L
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 13: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/13.jpg)
logo1
New Idea An Example Double Check
So Everything Remains As It Was
Time Domain (t) Transform domain (s)
OriginalDE & IVP
Algebraic equation forthe Laplace transform
-L
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 14: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/14.jpg)
logo1
New Idea An Example Double Check
So Everything Remains As It Was
Time Domain (t) Transform domain (s)
OriginalDE & IVP
Algebraic equation forthe Laplace transform
-L
Algebraic solution,partial fractions
?
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 15: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/15.jpg)
logo1
New Idea An Example Double Check
So Everything Remains As It Was
Time Domain (t) Transform domain (s)
OriginalDE & IVP
Algebraic equation forthe Laplace transform
Laplace transformof the solution
-L
Algebraic solution,partial fractions
?
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 16: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/16.jpg)
logo1
New Idea An Example Double Check
So Everything Remains As It Was
Time Domain (t) Transform domain (s)
OriginalDE & IVP
Algebraic equation forthe Laplace transform
Laplace transformof the solution
-
�
L
L −1
Algebraic solution,partial fractions
?
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 17: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/17.jpg)
logo1
New Idea An Example Double Check
So Everything Remains As It Was
Time Domain (t) Transform domain (s)
OriginalDE & IVP
Algebraic equation forthe Laplace transform
Laplace transformof the solutionSolution
-
�
L
L −1
Algebraic solution,partial fractions
?
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 18: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/18.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
1. Note that the second equation is not really a differentialequation.
2. This is not a problem. The transforms will work the sameway, ...
3. ... but the second equation also relates the initial values toeach other. So certain combinations of initial values willnot be possible.
4. The initial values in this example are allowed.5. The example itself is related to equations that come from
the analysis of two loop circuits. So systems such as thisone certainly arise in applications.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 19: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/19.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
1. Note that the second equation is not really a differentialequation.
2. This is not a problem. The transforms will work the sameway, ...
3. ... but the second equation also relates the initial values toeach other. So certain combinations of initial values willnot be possible.
4. The initial values in this example are allowed.5. The example itself is related to equations that come from
the analysis of two loop circuits. So systems such as thisone certainly arise in applications.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 20: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/20.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
1. Note that the second equation is not really a differentialequation.
2. This is not a problem. The transforms will work the sameway, ...
3. ... but the second equation also relates the initial values toeach other. So certain combinations of initial values willnot be possible.
4. The initial values in this example are allowed.5. The example itself is related to equations that come from
the analysis of two loop circuits. So systems such as thisone certainly arise in applications.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 21: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/21.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
1. Note that the second equation is not really a differentialequation.
2. This is not a problem. The transforms will work the sameway, ...
3. ... but the second equation also relates the initial values toeach other. So certain combinations of initial values willnot be possible.
4. The initial values in this example are allowed.5. The example itself is related to equations that come from
the analysis of two loop circuits. So systems such as thisone certainly arise in applications.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 22: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/22.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
1. Note that the second equation is not really a differentialequation.
2. This is not a problem. The transforms will work the sameway, ...
3. ... but the second equation also relates the initial values toeach other. So certain combinations of initial values willnot be possible.
4. The initial values in this example are allowed.
5. The example itself is related to equations that come fromthe analysis of two loop circuits. So systems such as thisone certainly arise in applications.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 23: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/23.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
1. Note that the second equation is not really a differentialequation.
2. This is not a problem. The transforms will work the sameway, ...
3. ... but the second equation also relates the initial values toeach other. So certain combinations of initial values willnot be possible.
4. The initial values in this example are allowed.5. The example itself is related to equations that come from
the analysis of two loop circuits. So systems such as thisone certainly arise in applications.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 24: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/24.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
6x+6y′+ y = 2e−t, x(0) = 12x− y = 0, y(0) = 2
6X +6sY−12+Y =2
s+12X−Y = 0 (×−3 and add)
Y(6s+4) =2
s+1+12 =
12s+14s+1
Y =12s+14
(s+1)(6s+4)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 25: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/25.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
6x+6y′+ y = 2e−t, x(0) = 12x− y = 0, y(0) = 2
6X +6sY−12+Y =2
s+12X−Y = 0 (×−3 and add)
Y(6s+4) =2
s+1+12 =
12s+14s+1
Y =12s+14
(s+1)(6s+4)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 26: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/26.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
6x+6y′+ y = 2e−t, x(0) = 12x− y = 0, y(0) = 2
6X +6sY−12+Y =2
s+1
2X−Y = 0 (×−3 and add)
Y(6s+4) =2
s+1+12 =
12s+14s+1
Y =12s+14
(s+1)(6s+4)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 27: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/27.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
6x+6y′+ y = 2e−t, x(0) = 12x− y = 0, y(0) = 2
6X +6sY−12+Y =2
s+12X−Y = 0
(×−3 and add)
Y(6s+4) =2
s+1+12 =
12s+14s+1
Y =12s+14
(s+1)(6s+4)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 28: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/28.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
6x+6y′+ y = 2e−t, x(0) = 12x− y = 0, y(0) = 2
6X +6sY−12+Y =2
s+12X−Y = 0 (×−3 and add)
Y(6s+4) =2
s+1+12 =
12s+14s+1
Y =12s+14
(s+1)(6s+4)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 29: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/29.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
6x+6y′+ y = 2e−t, x(0) = 12x− y = 0, y(0) = 2
6X +6sY−12+Y =2
s+12X−Y = 0 (×−3 and add)
Y
(6s+4) =2
s+1+12 =
12s+14s+1
Y =12s+14
(s+1)(6s+4)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 30: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/30.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
6x+6y′+ y = 2e−t, x(0) = 12x− y = 0, y(0) = 2
6X +6sY−12+Y =2
s+12X−Y = 0 (×−3 and add)
Y(6s
+4) =2
s+1+12 =
12s+14s+1
Y =12s+14
(s+1)(6s+4)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 31: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/31.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
6x+6y′+ y = 2e−t, x(0) = 12x− y = 0, y(0) = 2
6X +6sY−12+Y =2
s+12X−Y = 0 (×−3 and add)
Y(6s+4)
=2
s+1+12 =
12s+14s+1
Y =12s+14
(s+1)(6s+4)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 32: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/32.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
6x+6y′+ y = 2e−t, x(0) = 12x− y = 0, y(0) = 2
6X +6sY−12+Y =2
s+12X−Y = 0 (×−3 and add)
Y(6s+4) =2
s+1+12
=12s+14
s+1
Y =12s+14
(s+1)(6s+4)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 33: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/33.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
6x+6y′+ y = 2e−t, x(0) = 12x− y = 0, y(0) = 2
6X +6sY−12+Y =2
s+12X−Y = 0 (×−3 and add)
Y(6s+4) =2
s+1+12 =
12s+14s+1
Y =12s+14
(s+1)(6s+4)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 34: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/34.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
6x+6y′+ y = 2e−t, x(0) = 12x− y = 0, y(0) = 2
6X +6sY−12+Y =2
s+12X−Y = 0 (×−3 and add)
Y(6s+4) =2
s+1+12 =
12s+14s+1
Y =12s+14
(s+1)(6s+4)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 35: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/35.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
12s+14(s+1)(6s+4)
=A
s+1+
B6s+4
12s+14 = A(6s+4)+B(s+1)s =−1 : 2 = A(−2), A =−1
s =−23
: 6 = B(
13
), B = 18
Y(s) = − 1s+1
+18
6s+4=− 1
s+1+3
1s+ 2
3
y(t) = −e−t +3e−23 t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 36: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/36.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
12s+14(s+1)(6s+4)
=A
s+1+
B6s+4
12s+14 = A(6s+4)+B(s+1)s =−1 : 2 = A(−2), A =−1
s =−23
: 6 = B(
13
), B = 18
Y(s) = − 1s+1
+18
6s+4=− 1
s+1+3
1s+ 2
3
y(t) = −e−t +3e−23 t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 37: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/37.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
12s+14(s+1)(6s+4)
=A
s+1+
B6s+4
12s+14 = A(6s+4)+B(s+1)
s =−1 : 2 = A(−2), A =−1
s =−23
: 6 = B(
13
), B = 18
Y(s) = − 1s+1
+18
6s+4=− 1
s+1+3
1s+ 2
3
y(t) = −e−t +3e−23 t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 38: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/38.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
12s+14(s+1)(6s+4)
=A
s+1+
B6s+4
12s+14 = A(6s+4)+B(s+1)s =−1 :
2 = A(−2), A =−1
s =−23
: 6 = B(
13
), B = 18
Y(s) = − 1s+1
+18
6s+4=− 1
s+1+3
1s+ 2
3
y(t) = −e−t +3e−23 t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 39: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/39.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
12s+14(s+1)(6s+4)
=A
s+1+
B6s+4
12s+14 = A(6s+4)+B(s+1)s =−1 : 2
= A(−2), A =−1
s =−23
: 6 = B(
13
), B = 18
Y(s) = − 1s+1
+18
6s+4=− 1
s+1+3
1s+ 2
3
y(t) = −e−t +3e−23 t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 40: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/40.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
12s+14(s+1)(6s+4)
=A
s+1+
B6s+4
12s+14 = A(6s+4)+B(s+1)s =−1 : 2 = A(−2)
, A =−1
s =−23
: 6 = B(
13
), B = 18
Y(s) = − 1s+1
+18
6s+4=− 1
s+1+3
1s+ 2
3
y(t) = −e−t +3e−23 t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 41: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/41.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
12s+14(s+1)(6s+4)
=A
s+1+
B6s+4
12s+14 = A(6s+4)+B(s+1)s =−1 : 2 = A(−2), A =−1
s =−23
: 6 = B(
13
), B = 18
Y(s) = − 1s+1
+18
6s+4=− 1
s+1+3
1s+ 2
3
y(t) = −e−t +3e−23 t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 42: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/42.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
12s+14(s+1)(6s+4)
=A
s+1+
B6s+4
12s+14 = A(6s+4)+B(s+1)s =−1 : 2 = A(−2), A =−1
s =−23
:
6 = B(
13
), B = 18
Y(s) = − 1s+1
+18
6s+4=− 1
s+1+3
1s+ 2
3
y(t) = −e−t +3e−23 t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 43: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/43.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
12s+14(s+1)(6s+4)
=A
s+1+
B6s+4
12s+14 = A(6s+4)+B(s+1)s =−1 : 2 = A(−2), A =−1
s =−23
: 6
= B(
13
), B = 18
Y(s) = − 1s+1
+18
6s+4=− 1
s+1+3
1s+ 2
3
y(t) = −e−t +3e−23 t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 44: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/44.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
12s+14(s+1)(6s+4)
=A
s+1+
B6s+4
12s+14 = A(6s+4)+B(s+1)s =−1 : 2 = A(−2), A =−1
s =−23
: 6 = B(
13
)
, B = 18
Y(s) = − 1s+1
+18
6s+4=− 1
s+1+3
1s+ 2
3
y(t) = −e−t +3e−23 t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 45: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/45.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
12s+14(s+1)(6s+4)
=A
s+1+
B6s+4
12s+14 = A(6s+4)+B(s+1)s =−1 : 2 = A(−2), A =−1
s =−23
: 6 = B(
13
), B = 18
Y(s) = − 1s+1
+18
6s+4=− 1
s+1+3
1s+ 2
3
y(t) = −e−t +3e−23 t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 46: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/46.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
12s+14(s+1)(6s+4)
=A
s+1+
B6s+4
12s+14 = A(6s+4)+B(s+1)s =−1 : 2 = A(−2), A =−1
s =−23
: 6 = B(
13
), B = 18
Y(s) = − 1s+1
+18
6s+4
=− 1s+1
+31
s+ 23
y(t) = −e−t +3e−23 t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 47: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/47.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
12s+14(s+1)(6s+4)
=A
s+1+
B6s+4
12s+14 = A(6s+4)+B(s+1)s =−1 : 2 = A(−2), A =−1
s =−23
: 6 = B(
13
), B = 18
Y(s) = − 1s+1
+18
6s+4=− 1
s+1+3
1s+ 2
3
y(t) = −e−t +3e−23 t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 48: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/48.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
12s+14(s+1)(6s+4)
=A
s+1+
B6s+4
12s+14 = A(6s+4)+B(s+1)s =−1 : 2 = A(−2), A =−1
s =−23
: 6 = B(
13
), B = 18
Y(s) = − 1s+1
+18
6s+4=− 1
s+1+3
1s+ 2
3
y(t) = −e−t +3e−23 t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 49: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/49.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
2X−Y = 0
X =12
Y
=12
[− 1
s+1+3
1s+ 2
3
]= −1
21
s+1+
32
1s+ 2
3
x(t) = −12
e−t +32
e−23 t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 50: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/50.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
2X−Y = 0
X =12
Y
=12
[− 1
s+1+3
1s+ 2
3
]= −1
21
s+1+
32
1s+ 2
3
x(t) = −12
e−t +32
e−23 t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 51: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/51.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
2X−Y = 0
X =12
Y
=12
[− 1
s+1+3
1s+ 2
3
]= −1
21
s+1+
32
1s+ 2
3
x(t) = −12
e−t +32
e−23 t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 52: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/52.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
2X−Y = 0
X =12
Y
=12
[− 1
s+1+3
1s+ 2
3
]
= −12
1s+1
+32
1s+ 2
3
x(t) = −12
e−t +32
e−23 t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 53: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/53.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
2X−Y = 0
X =12
Y
=12
[− 1
s+1+3
1s+ 2
3
]= −1
21
s+1+
32
1s+ 2
3
x(t) = −12
e−t +32
e−23 t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 54: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/54.jpg)
logo1
New Idea An Example Double Check
Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2
2X−Y = 0
X =12
Y
=12
[− 1
s+1+3
1s+ 2
3
]= −1
21
s+1+
32
1s+ 2
3
x(t) = −12
e−t +32
e−23 t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 55: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/55.jpg)
logo1
New Idea An Example Double Check
Does x(t) =−12
e−t +32
e−23 t, y(t) =−e−t +3e−
23 t
Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2
Initial values: Look at x and y!2x− y = 0: Same as x =
y2
. Look at x and y!
6x+6y′+ y = 6[−1
2e−t +
32
e−23 t]
+6[e−t−2e−
23 t]
+[−e−t +3e−
23 t]
= e−t (−3+6−1) + e−23 t (9−12+3)
= 2e−t √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 56: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/56.jpg)
logo1
New Idea An Example Double Check
Does x(t) =−12
e−t +32
e−23 t, y(t) =−e−t +3e−
23 t
Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2
Initial values:
Look at x and y!2x− y = 0: Same as x =
y2
. Look at x and y!
6x+6y′+ y = 6[−1
2e−t +
32
e−23 t]
+6[e−t−2e−
23 t]
+[−e−t +3e−
23 t]
= e−t (−3+6−1) + e−23 t (9−12+3)
= 2e−t √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 57: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/57.jpg)
logo1
New Idea An Example Double Check
Does x(t) =−12
e−t +32
e−23 t, y(t) =−e−t +3e−
23 t
Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2
Initial values: Look at x and y!
2x− y = 0: Same as x =y2
. Look at x and y!
6x+6y′+ y = 6[−1
2e−t +
32
e−23 t]
+6[e−t−2e−
23 t]
+[−e−t +3e−
23 t]
= e−t (−3+6−1) + e−23 t (9−12+3)
= 2e−t √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 58: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/58.jpg)
logo1
New Idea An Example Double Check
Does x(t) =−12
e−t +32
e−23 t, y(t) =−e−t +3e−
23 t
Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2
Initial values: Look at x and y!2x− y = 0:
Same as x =y2
. Look at x and y!
6x+6y′+ y = 6[−1
2e−t +
32
e−23 t]
+6[e−t−2e−
23 t]
+[−e−t +3e−
23 t]
= e−t (−3+6−1) + e−23 t (9−12+3)
= 2e−t √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 59: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/59.jpg)
logo1
New Idea An Example Double Check
Does x(t) =−12
e−t +32
e−23 t, y(t) =−e−t +3e−
23 t
Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2
Initial values: Look at x and y!2x− y = 0: Same as x =
y2
.
Look at x and y!
6x+6y′+ y = 6[−1
2e−t +
32
e−23 t]
+6[e−t−2e−
23 t]
+[−e−t +3e−
23 t]
= e−t (−3+6−1) + e−23 t (9−12+3)
= 2e−t √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 60: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/60.jpg)
logo1
New Idea An Example Double Check
Does x(t) =−12
e−t +32
e−23 t, y(t) =−e−t +3e−
23 t
Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2
Initial values: Look at x and y!2x− y = 0: Same as x =
y2
. Look at x and y!
6x+6y′+ y = 6[−1
2e−t +
32
e−23 t]
+6[e−t−2e−
23 t]
+[−e−t +3e−
23 t]
= e−t (−3+6−1) + e−23 t (9−12+3)
= 2e−t √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 61: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/61.jpg)
logo1
New Idea An Example Double Check
Does x(t) =−12
e−t +32
e−23 t, y(t) =−e−t +3e−
23 t
Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2
Initial values: Look at x and y!2x− y = 0: Same as x =
y2
. Look at x and y!
6x+6y′+ y =
6[−1
2e−t +
32
e−23 t]
+6[e−t−2e−
23 t]
+[−e−t +3e−
23 t]
= e−t (−3+6−1) + e−23 t (9−12+3)
= 2e−t √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 62: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/62.jpg)
logo1
New Idea An Example Double Check
Does x(t) =−12
e−t +32
e−23 t, y(t) =−e−t +3e−
23 t
Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2
Initial values: Look at x and y!2x− y = 0: Same as x =
y2
. Look at x and y!
6x+6y′+ y = 6[−1
2e−t +
32
e−23 t]
+6[e−t−2e−
23 t]
+[−e−t +3e−
23 t]
= e−t (−3+6−1) + e−23 t (9−12+3)
= 2e−t √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 63: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/63.jpg)
logo1
New Idea An Example Double Check
Does x(t) =−12
e−t +32
e−23 t, y(t) =−e−t +3e−
23 t
Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2
Initial values: Look at x and y!2x− y = 0: Same as x =
y2
. Look at x and y!
6x+6y′+ y = 6[−1
2e−t +
32
e−23 t]
+6[e−t−2e−
23 t]
+[−e−t +3e−
23 t]
= e−t (−3+6−1) + e−23 t (9−12+3)
= 2e−t √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 64: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/64.jpg)
logo1
New Idea An Example Double Check
Does x(t) =−12
e−t +32
e−23 t, y(t) =−e−t +3e−
23 t
Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2
Initial values: Look at x and y!2x− y = 0: Same as x =
y2
. Look at x and y!
6x+6y′+ y = 6[−1
2e−t +
32
e−23 t]
+6[e−t−2e−
23 t]
+[−e−t +3e−
23 t]
= e−t (−3+6−1) + e−23 t (9−12+3)
= 2e−t √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 65: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/65.jpg)
logo1
New Idea An Example Double Check
Does x(t) =−12
e−t +32
e−23 t, y(t) =−e−t +3e−
23 t
Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2
Initial values: Look at x and y!2x− y = 0: Same as x =
y2
. Look at x and y!
6x+6y′+ y = 6[−1
2e−t +
32
e−23 t]
+6[e−t−2e−
23 t]
+[−e−t +3e−
23 t]
= e−t
(−3+6−1) + e−23 t (9−12+3)
= 2e−t √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 66: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/66.jpg)
logo1
New Idea An Example Double Check
Does x(t) =−12
e−t +32
e−23 t, y(t) =−e−t +3e−
23 t
Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2
Initial values: Look at x and y!2x− y = 0: Same as x =
y2
. Look at x and y!
6x+6y′+ y = 6[−1
2e−t +
32
e−23 t]
+6[e−t−2e−
23 t]
+[−e−t +3e−
23 t]
= e−t (−3
+6−1) + e−23 t (9−12+3)
= 2e−t √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 67: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/67.jpg)
logo1
New Idea An Example Double Check
Does x(t) =−12
e−t +32
e−23 t, y(t) =−e−t +3e−
23 t
Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2
Initial values: Look at x and y!2x− y = 0: Same as x =
y2
. Look at x and y!
6x+6y′+ y = 6[−1
2e−t +
32
e−23 t]
+6[e−t−2e−
23 t]
+[−e−t +3e−
23 t]
= e−t (−3+6
−1) + e−23 t (9−12+3)
= 2e−t √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 68: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/68.jpg)
logo1
New Idea An Example Double Check
Does x(t) =−12
e−t +32
e−23 t, y(t) =−e−t +3e−
23 t
Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2
Initial values: Look at x and y!2x− y = 0: Same as x =
y2
. Look at x and y!
6x+6y′+ y = 6[−1
2e−t +
32
e−23 t]
+6[e−t−2e−
23 t]
+[−e−t +3e−
23 t]
= e−t (−3+6−1)
+ e−23 t (9−12+3)
= 2e−t √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 69: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/69.jpg)
logo1
New Idea An Example Double Check
Does x(t) =−12
e−t +32
e−23 t, y(t) =−e−t +3e−
23 t
Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2
Initial values: Look at x and y!2x− y = 0: Same as x =
y2
. Look at x and y!
6x+6y′+ y = 6[−1
2e−t +
32
e−23 t]
+6[e−t−2e−
23 t]
+[−e−t +3e−
23 t]
= e−t (−3+6−1) + e−23 t
(9−12+3)= 2e−t √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 70: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/70.jpg)
logo1
New Idea An Example Double Check
Does x(t) =−12
e−t +32
e−23 t, y(t) =−e−t +3e−
23 t
Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2
Initial values: Look at x and y!2x− y = 0: Same as x =
y2
. Look at x and y!
6x+6y′+ y = 6[−1
2e−t +
32
e−23 t]
+6[e−t−2e−
23 t]
+[−e−t +3e−
23 t]
= e−t (−3+6−1) + e−23 t (9
−12+3)= 2e−t √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 71: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/71.jpg)
logo1
New Idea An Example Double Check
Does x(t) =−12
e−t +32
e−23 t, y(t) =−e−t +3e−
23 t
Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2
Initial values: Look at x and y!2x− y = 0: Same as x =
y2
. Look at x and y!
6x+6y′+ y = 6[−1
2e−t +
32
e−23 t]
+6[e−t−2e−
23 t]
+[−e−t +3e−
23 t]
= e−t (−3+6−1) + e−23 t (9−12
+3)= 2e−t √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 72: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/72.jpg)
logo1
New Idea An Example Double Check
Does x(t) =−12
e−t +32
e−23 t, y(t) =−e−t +3e−
23 t
Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2
Initial values: Look at x and y!2x− y = 0: Same as x =
y2
. Look at x and y!
6x+6y′+ y = 6[−1
2e−t +
32
e−23 t]
+6[e−t−2e−
23 t]
+[−e−t +3e−
23 t]
= e−t (−3+6−1) + e−23 t (9−12+3)
= 2e−t √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 73: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/73.jpg)
logo1
New Idea An Example Double Check
Does x(t) =−12
e−t +32
e−23 t, y(t) =−e−t +3e−
23 t
Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2
Initial values: Look at x and y!2x− y = 0: Same as x =
y2
. Look at x and y!
6x+6y′+ y = 6[−1
2e−t +
32
e−23 t]
+6[e−t−2e−
23 t]
+[−e−t +3e−
23 t]
= e−t (−3+6−1) + e−23 t (9−12+3)
= 2e−t
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations
![Page 74: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨](https://reader030.fdocuments.in/reader030/viewer/2022040200/5e347594a5a3d45eeb3bde71/html5/thumbnails/74.jpg)
logo1
New Idea An Example Double Check
Does x(t) =−12
e−t +32
e−23 t, y(t) =−e−t +3e−
23 t
Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2
Initial values: Look at x and y!2x− y = 0: Same as x =
y2
. Look at x and y!
6x+6y′+ y = 6[−1
2e−t +
32
e−23 t]
+6[e−t−2e−
23 t]
+[−e−t +3e−
23 t]
= e−t (−3+6−1) + e−23 t (9−12+3)
= 2e−t √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms for Systems of Differential Equations