Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) •...
Transcript of Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) •...
![Page 1: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/1.jpg)
Today
• Non-homogeneous systems of ODEs
• Non-homogeneous two-tank example
• Intro to Laplace transforms
![Page 2: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/2.jpg)
Nonhomogeneous system of DEs
• How do you solve the equation
x�(t) = Ax(t) + b ?
![Page 3: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/3.jpg)
Nonhomogeneous system of DEs
• How do you solve the equation
• Define the linear operator
x�(t) = Ax(t) + b
L[x] = x�(t)−Ax(t)
?
![Page 4: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/4.jpg)
Nonhomogeneous system of DEs
• How do you solve the equation
• Define the linear operator
• The equation above can be written as
x�(t) = Ax(t) + b
L[x] = x�(t)−Ax(t)
L[x] = b
?
![Page 5: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/5.jpg)
Nonhomogeneous system of DEs
• How do you solve the equation
• Define the linear operator
• The equation above can be written as
• As for 2nd order equations, solve homogeneous eqn first,
x�(t) = Ax(t) + b
L[x] = x�(t)−Ax(t)
L[x] = b
?
L[x] = 0
![Page 6: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/6.jpg)
Nonhomogeneous system of DEs
• How do you solve the equation
• Define the linear operator
• The equation above can be written as
• As for 2nd order equations, solve homogeneous eqn first,
• then Method of Undetermined Coefficients...
x�(t) = Ax(t) + b
L[x] = x�(t)−Ax(t)
L[x] = b
?
L[x] = 0
![Page 7: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/7.jpg)
Nonhomogeneous system of DEs
• For the equation,
which of the following is a suitable guess (in the sense of MUC)?
x�(t) = Ax(t) + b
(A)
(B)
(C)
(D)
(E) Huh?
xp = cb
xp = v
xp = tv
xp = tu + v
![Page 8: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/8.jpg)
Nonhomogeneous system of DEs
• For the equation,
which of the following is a suitable guess (in the sense of MUC)?
x�(t) = Ax(t) + b
(A)
(B)
(C)
(D)
(E) Huh?
xp = cb
xp = v
xp = tv
xp = tu + v
-- works only when b happens to be an eigenvector associated with a non-zero eigenvalue; not really worth trying.
![Page 9: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/9.jpg)
Nonhomogeneous system of DEs
• For the equation,
which of the following is a suitable guess (in the sense of MUC)?
x�(t) = Ax(t) + b
(A)
(B)
(C)
(D)
(E) Huh?
xp = cb
xp = v
xp = tv
xp = tu + v
-- works only when b happens to be an eigenvector associated with a non-zero eigenvalue; not really worth trying.
-- works when b is in the range of A (which is to say often so try this first).
![Page 10: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/10.jpg)
Nonhomogeneous system of DEs
• For the equation,
which of the following is a suitable guess (in the sense of MUC)?
x�(t) = Ax(t) + b
(A)
(B)
(C)
(D)
(E) Huh?
xp = cb
xp = v
xp = tv
xp = tu + v
-- works only when b happens to be an eigenvector associated with a non-zero eigenvalue; not really worth trying.
-- works when b is in the range of A (which is to say often so try this first).
-- works when (B) doesn’t and b happens to be in the nullspace of A.
![Page 11: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/11.jpg)
Nonhomogeneous system of DEs
• For the equation,
which of the following is a suitable guess (in the sense of MUC)?
x�(t) = Ax(t) + b
(A)
(B)
(C)
(D)
(E) Huh?
xp = cb
xp = v
xp = tv
xp = tu + v
-- works only when b happens to be an eigenvector associated with a non-zero eigenvalue; not really worth trying.
-- works when b is in the range of A (which is to say often so try this first).
-- works when (B) doesn’t and b happens to be in the nullspace of A.
-- works when (B) and (C) don’t with one exception but is beyond the scope of this course.
![Page 12: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/12.jpg)
Nonhomogeneous system of DEs - example
• Salt water flows into a tank holding 10 L of water at a rate of 1 L/min with a concentration of 200 g/L. The well-mixed solution flows from that tank into a tank holding 5 L through a pipe at 3 L/min. Another pipe takes the solution in the second tank back into the first at a rate of 2 L/min. Finally, solution drains out of the second tank at a rate of 1 L/min.
• Write down a system of equations in matrix form for the mass of salt in each tank.
![Page 13: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/13.jpg)
Nonhomogeneous system of DEs - example
• Salt water flows into a tank holding 10 L of water at a rate of 1 L/min with a concentration of 200 g/L. The well-mixed solution flows from that tank into a tank holding 5 L through a pipe at 3 L/min. Another pipe takes the solution in the second tank back into the first at a rate of 2 L/min. Finally, solution drains out of the second tank at a rate of 1 L/min.
• Write down a system of equations in matrix form for the mass of salt in each tank.
�m1
m2
��=
− 310
25
310
−35
�m1
m2
�+
�2000
�
![Page 14: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/14.jpg)
Nonhomogeneous case - example
• Salt water flows into a tank holding 10 L of water at a rate of 1 L/min with a concentration of 200 g/L. The well-mixed solution flows from that tank into a tank holding 5 L through a pipe at 3 L/min. Another pipe takes the solution in the second tank back into the first at a rate of 2 L/min. Finally, solution drains out of the second tank at a rate of 1 L/min.
• Find the eigenvalues and the long term (steady state) solution.
![Page 15: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/15.jpg)
Nonhomogeneous case - example
• Salt water flows into a tank holding 10 L of water at a rate of 1 L/min with a concentration of 200 g/L. The well-mixed solution flows from that tank into a tank holding 5 L through a pipe at 3 L/min. Another pipe takes the solution in the second tank back into the first at a rate of 2 L/min. Finally, solution drains out of the second tank at a rate of 1 L/min.
• Find the eigenvalues and the long term (steady state) solution.
�m1
m2
��=
− 310
25
310
−35
�m1
m2
�+
�2000
�
![Page 16: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/16.jpg)
Nonhomogeneous case - example
• Salt water flows into a tank holding 10 L of water at a rate of 1 L/min with a concentration of 200 g/L. The well-mixed solution flows from that tank into a tank holding 5 L through a pipe at 3 L/min. Another pipe takes the solution in the second tank back into the first at a rate of 2 L/min. Finally, solution drains out of the second tank at a rate of 1 L/min.
• Find the eigenvalues and the long term (steady state) solution.
�m1
m2
��=
− 310
25
310
−35
�m1
m2
�+
�2000
�
trA = − 910
detA =950− 6
50=
350
4 detA =1250
(trA)2 =81100 Both evalues
negative!
![Page 17: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/17.jpg)
Nonhomogeneous case - example
�m1
m2
��=
− 310
25
310
−35
�m1
m2
�+
�2000
�
Both evalues negative!
![Page 18: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/18.jpg)
Nonhomogeneous case - example
�m1
m2
��=
− 310
25
310
−35
�m1
m2
�+
�2000
�
mh(t) = C1eλ1tv1 + C2e
λ2tv2
�λ1,2 = − 9
20±√
5720
�Both evalues
negative!
![Page 19: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/19.jpg)
Nonhomogeneous case - example
�m1
m2
��=
− 310
25
310
−35
�m1
m2
�+
�2000
�
mh(t) = C1eλ1tv1 + C2e
λ2tv2
�λ1,2 = − 9
20±√
5720
�Both evalues
negative!
mp(t) =
![Page 20: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/20.jpg)
Nonhomogeneous case - example
�m1
m2
��=
− 310
25
310
−35
�m1
m2
�+
�2000
�
mh(t) = C1eλ1tv1 + C2e
λ2tv2
�λ1,2 = − 9
20±√
5720
�Both evalues
negative!
mp(t) = w =�
w1
w2
�
![Page 21: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/21.jpg)
Nonhomogeneous case - example
�m1
m2
��=
− 310
25
310
−35
�m1
m2
�+
�2000
�
mh(t) = C1eλ1tv1 + C2e
λ2tv2
�λ1,2 = − 9
20±√
5720
�
0 = Aw +�
2000
�
Both evalues negative!
mp(t) = w =�
w1
w2
�
![Page 22: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/22.jpg)
Nonhomogeneous case - example
�m1
m2
��=
− 310
25
310
−35
�m1
m2
�+
�2000
�
mh(t) = C1eλ1tv1 + C2e
λ2tv2
�λ1,2 = − 9
20±√
5720
�
0 = Aw +�
2000
�
Both evalues negative!
mp(t) = w =�
w1
w2
�
![Page 23: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/23.jpg)
Nonhomogeneous case - example
�m1
m2
��=
− 310
25
310
−35
�m1
m2
�+
�2000
�
mh(t) = C1eλ1tv1 + C2e
λ2tv2
�λ1,2 = − 9
20±√
5720
�
0 = Aw +�
2000
�→ Aw = −
�2000
�
Both evalues negative!
mp(t) = w =�
w1
w2
�
![Page 24: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/24.jpg)
Nonhomogeneous case - example
�m1
m2
��=
− 310
25
310
−35
�m1
m2
�+
�2000
�
mh(t) = C1eλ1tv1 + C2e
λ2tv2
�λ1,2 = − 9
20±√
5720
�
0 = Aw +�
2000
�→ Aw = −
�2000
�→ w =
�20001000
�
Both evalues negative!
mp(t) = w =�
w1
w2
�
![Page 25: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/25.jpg)
Nonhomogeneous case - example
�m1
m2
��=
− 310
25
310
−35
�m1
m2
�+
�2000
�
mh(t) = C1eλ1tv1 + C2e
λ2tv2
�λ1,2 = − 9
20±√
5720
�
0 = Aw +�
2000
�→ Aw = −
�2000
�→ w =
�20001000
�
Both evalues negative!
mp(t) = w =�
w1
w2
�
m(t) = C1eλ1tv1 + C2e
λ2tv2 +�
20001000
�
![Page 26: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/26.jpg)
Nonhomogeneous case - example
• A “Method of undetermined coefficients” similar to what we saw for second order equations can be used for systems.
• For this course, I’ll only test you on constant nonhomogeneous terms (like the previous example).
![Page 27: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/27.jpg)
Laplace transforms - intro (6.1)
• Motivation for Laplace transforms:
![Page 28: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/28.jpg)
Laplace transforms - intro (6.1)
• Motivation for Laplace transforms:
• We know how to solve when is polynomial, exponential, trig.
ay�� + by� + cy = g(t) g(t)
![Page 29: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/29.jpg)
Laplace transforms - intro (6.1)
• Motivation for Laplace transforms:
• We know how to solve when is polynomial, exponential, trig.
• In applications, is often “piece-wise continuous” meaning that it consists of a finite number of pieces with jump discontinuities in between. For example,
ay�� + by� + cy = g(t) g(t)
g(t)
g(t) =�
sin(ωt) 0 < t < 10,0 t ≥ 10.
![Page 30: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/30.jpg)
Laplace transforms - intro (6.1)
• Motivation for Laplace transforms:
• We know how to solve when is polynomial, exponential, trig.
• In applications, is often “piece-wise continuous” meaning that it consists of a finite number of pieces with jump discontinuities in between. For example,
• These can be handled by previous techniques (modified) but it isn’t pretty (solve from t=0 to t=10, use y(10) as the IC for a new problem starting at t=10).
ay�� + by� + cy = g(t) g(t)
g(t)
g(t) =�
sin(ωt) 0 < t < 10,0 t ≥ 10.
![Page 31: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/31.jpg)
Laplace transforms - intro (6.1)
![Page 32: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/32.jpg)
Laplace transforms - intro (6.1)
• Motivation for Laplace transforms - example RLC circuit
![Page 33: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/33.jpg)
Laplace transforms - intro (6.1)
• Motivation for Laplace transforms - example RLC circuit
• Resistor, inductor and capacitor in series
I ��(t) +R
LI �(t) +
1LC
I(t) = v(t)
![Page 34: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/34.jpg)
Laplace transforms - intro (6.1)
• Motivation for Laplace transforms - example RLC circuit
• Resistor, inductor and capacitor in series
• If v(t) comes from radio waves then and the circuit is called a radio receiver.
I ��(t) +R
LI �(t) +
1LC
I(t) = v(t)
v(t) = A cos(ωt)
![Page 35: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/35.jpg)
Laplace transforms - intro (6.1)
• Motivation for Laplace transforms - example RLC circuit
• Resistor, inductor and capacitor in series
• If v(t) comes from radio waves then and the circuit is called a radio receiver.
• For , the circuit has a switch that gets
flipped at t=10.
I ��(t) +R
LI �(t) +
1LC
I(t) = v(t)
v(t) = A cos(ωt)
v(t) =�
1 0 < t < 100 t ≥ 10
![Page 36: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/36.jpg)
Laplace transforms - intro (6.1)
• Instead of not-so-pretty techniques, we use Laplace transforms.
• Idea:
![Page 37: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/37.jpg)
Laplace transforms - intro (6.1)
• Instead of not-so-pretty techniques, we use Laplace transforms.
• Idea:
Unknown y(t) that satisfies some ODE
Found y(t)solve ODE
![Page 38: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/38.jpg)
Laplace transforms - intro (6.1)
• Instead of not-so-pretty techniques, we use Laplace transforms.
• Idea:
Unknown y(t) that satisfies some ODE
Found y(t)solve ODE
Unknown Y(s) that satisfies an algebraic
equation
Transform y(t) and the ODE
![Page 39: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/39.jpg)
Laplace transforms - intro (6.1)
• Instead of not-so-pretty techniques, we use Laplace transforms.
• Idea:
Unknown y(t) that satisfies some ODE
Found y(t)solve ODE
Found Y(s)solve algebraic eqUnknown Y(s) that
satisfies an algebraic equation
Transform y(t) and the ODE
![Page 40: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/40.jpg)
Laplace transforms - intro (6.1)
• Instead of not-so-pretty techniques, we use Laplace transforms.
• Idea:
Unknown y(t) that satisfies some ODE
Found y(t)solve ODE
Found Y(s)solve algebraic eqUnknown Y(s) that
satisfies an algebraic equation
Transform y(t) and the ODE
Invert the transform
![Page 41: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/41.jpg)
Laplace transforms - intro (6.1)
• Instead of not-so-pretty techniques, we use Laplace transforms.
• Idea:
Unknown y(t) that satisfies some ODE
Found y(t)solve ODE
Found Y(s)solve algebraic eqUnknown Y(s) that
satisfies an algebraic equation
Transform y(t) and the ODE
Invert the transform
• Laplace transform of y(t): L{y(t)} = Y (s) =� ∞
0e−sty(t) dt
![Page 42: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/42.jpg)
Laplace transforms - examples (6.1)
• What is the Laplace transform of ?y(t) = 3
L{y(t)} = Y (s) =� ∞
0e−st3 dt
![Page 43: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/43.jpg)
Laplace transforms - examples (6.1)
• What is the Laplace transform of ?y(t) = 3
L{y(t)} = Y (s) =� ∞
0e−st3 dt
= −3se−st
����∞
0
= limA→∞
−3se−st
����A
0
= −3s
�lim
A→∞e−sA − 1
�
=3s
provided and does not
exist otherwise.s > 0
![Page 44: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/44.jpg)
Laplace transforms - examples (6.1)
• What is the Laplace transform of ?y(t) = 3
L{y(t)} = Y (s) =� ∞
0e−st3 dt
=3s
provided and does not
exist otherwise.s > 0
![Page 45: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/45.jpg)
Laplace transforms - examples (6.1)
• What is the Laplace transform of ?y(t) = 3
L{y(t)} = Y (s) =� ∞
0e−st3 dt
=3s
provided and does not
exist otherwise.s > 0
3
![Page 46: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/46.jpg)
Laplace transforms - examples (6.1)
• What is the Laplace transform of ?y(t) = 3
L{y(t)} = Y (s) =� ∞
0e−st3 dt
=3s
provided and does not
exist otherwise.s > 0
3
![Page 47: Non-homogeneous systems of ODEs • Non-homogeneous two …Laplace transforms - intro (6.1) • Motivation for Laplace transforms: • We know how to solve when is polynomial, exponential,](https://reader030.fdocuments.in/reader030/viewer/2022041112/5f16a8e07bd9b85139796e6d/html5/thumbnails/47.jpg)
Laplace transforms - examples (6.1)
• What is the Laplace transform of ?y(t) = 3
L{y(t)} = Y (s) =� ∞
0e−st3 dt
=3s
provided and does not
exist otherwise.s > 0
33/s