Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order...
Transcript of Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order...
![Page 1: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/1.jpg)
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Overview An Example Double Check
Linear First Order Differential Equations
Bernd Schroder
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 2: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/2.jpg)
logo1
Overview An Example Double Check
What are Linear First Order DifferentialEquations?
1. A linear first order differential equation is of the formy′ +p(x)y = q(x).
2. Recognizing linear first order differential equationsrequires some pattern recognition.
3. To solve a linear first order differential equation, we turnthe left side into the derivative of a product.3.1 Compute the integrating factor µ(x) = e
∫p(x) dx,
3.2 Multiply the equation by the integrating factor,note that the left side e
∫p(x) dxy′ + e
∫p(x) dxp(x)y
is the derivative of the product e∫
p(x) dxy,3.3 Integrate both sides, solve for y.
That’s it.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 3: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/3.jpg)
logo1
Overview An Example Double Check
What are Linear First Order DifferentialEquations?
1. A linear first order differential equation is of the formy′ +p(x)y = q(x).
2. Recognizing linear first order differential equationsrequires some pattern recognition.
3. To solve a linear first order differential equation, we turnthe left side into the derivative of a product.3.1 Compute the integrating factor µ(x) = e
∫p(x) dx,
3.2 Multiply the equation by the integrating factor,note that the left side e
∫p(x) dxy′ + e
∫p(x) dxp(x)y
is the derivative of the product e∫
p(x) dxy,3.3 Integrate both sides, solve for y.
That’s it.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 4: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/4.jpg)
logo1
Overview An Example Double Check
What are Linear First Order DifferentialEquations?
1. A linear first order differential equation is of the formy′ +p(x)y = q(x).
2. Recognizing linear first order differential equationsrequires some pattern recognition.
3. To solve a linear first order differential equation, we turnthe left side into the derivative of a product.3.1 Compute the integrating factor µ(x) = e
∫p(x) dx,
3.2 Multiply the equation by the integrating factor,note that the left side e
∫p(x) dxy′ + e
∫p(x) dxp(x)y
is the derivative of the product e∫
p(x) dxy,3.3 Integrate both sides, solve for y.
That’s it.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 5: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/5.jpg)
logo1
Overview An Example Double Check
What are Linear First Order DifferentialEquations?
1. A linear first order differential equation is of the formy′ +p(x)y = q(x).
2. Recognizing linear first order differential equationsrequires some pattern recognition.
3. To solve a linear first order differential equation, we turnthe left side into the derivative of a product.
3.1 Compute the integrating factor µ(x) = e∫
p(x) dx,3.2 Multiply the equation by the integrating factor,
note that the left side e∫
p(x) dxy′ + e∫
p(x) dxp(x)yis the derivative of the product e
∫p(x) dxy,
3.3 Integrate both sides, solve for y.
That’s it.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 6: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/6.jpg)
logo1
Overview An Example Double Check
What are Linear First Order DifferentialEquations?
1. A linear first order differential equation is of the formy′ +p(x)y = q(x).
2. Recognizing linear first order differential equationsrequires some pattern recognition.
3. To solve a linear first order differential equation, we turnthe left side into the derivative of a product.3.1 Compute the integrating factor µ(x) = e
∫p(x) dx,
3.2 Multiply the equation by the integrating factor,note that the left side e
∫p(x) dxy′ + e
∫p(x) dxp(x)y
is the derivative of the product e∫
p(x) dxy,3.3 Integrate both sides, solve for y.
That’s it.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 7: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/7.jpg)
logo1
Overview An Example Double Check
What are Linear First Order DifferentialEquations?
1. A linear first order differential equation is of the formy′ +p(x)y = q(x).
2. Recognizing linear first order differential equationsrequires some pattern recognition.
3. To solve a linear first order differential equation, we turnthe left side into the derivative of a product.3.1 Compute the integrating factor µ(x) = e
∫p(x) dx,
3.2 Multiply the equation by the integrating factor,
note that the left side e∫
p(x) dxy′ + e∫
p(x) dxp(x)yis the derivative of the product e
∫p(x) dxy,
3.3 Integrate both sides, solve for y.
That’s it.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 8: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/8.jpg)
logo1
Overview An Example Double Check
What are Linear First Order DifferentialEquations?
1. A linear first order differential equation is of the formy′ +p(x)y = q(x).
2. Recognizing linear first order differential equationsrequires some pattern recognition.
3. To solve a linear first order differential equation, we turnthe left side into the derivative of a product.3.1 Compute the integrating factor µ(x) = e
∫p(x) dx,
3.2 Multiply the equation by the integrating factor,note that the left side e
∫p(x) dxy′ + e
∫p(x) dxp(x)y
is the derivative of the product e∫
p(x) dxy,
3.3 Integrate both sides, solve for y.
That’s it.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 9: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/9.jpg)
logo1
Overview An Example Double Check
What are Linear First Order DifferentialEquations?
1. A linear first order differential equation is of the formy′ +p(x)y = q(x).
2. Recognizing linear first order differential equationsrequires some pattern recognition.
3. To solve a linear first order differential equation, we turnthe left side into the derivative of a product.3.1 Compute the integrating factor µ(x) = e
∫p(x) dx,
3.2 Multiply the equation by the integrating factor,note that the left side e
∫p(x) dxy′ + e
∫p(x) dxp(x)y
is the derivative of the product e∫
p(x) dxy,3.3 Integrate both sides, solve for y.
That’s it.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 10: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/10.jpg)
logo1
Overview An Example Double Check
What are Linear First Order DifferentialEquations?
1. A linear first order differential equation is of the formy′ +p(x)y = q(x).
2. Recognizing linear first order differential equationsrequires some pattern recognition.
3. To solve a linear first order differential equation, we turnthe left side into the derivative of a product.3.1 Compute the integrating factor µ(x) = e
∫p(x) dx,
3.2 Multiply the equation by the integrating factor,note that the left side e
∫p(x) dxy′ + e
∫p(x) dxp(x)y
is the derivative of the product e∫
p(x) dxy,3.3 Integrate both sides, solve for y.
That’s it.Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 11: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/11.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.
Integrating factor:∫p(x) dx =
∫ cos(x)sin(x)
dx u := sin(x),dudx
= cos(x),
dx =du
cos(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 12: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/12.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =
∫ cos(x)sin(x)
dx u := sin(x),dudx
= cos(x),
dx =du
cos(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 13: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/13.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx
u := sin(x),dudx
= cos(x),
dx =du
cos(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 14: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/14.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx u := sin(x),
dudx
= cos(x),
dx =du
cos(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 15: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/15.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx u := sin(x),
dudx
= cos(x),
dx =du
cos(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 16: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/16.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx u := sin(x),
dudx
= cos(x),
dx =du
cos(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 17: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/17.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx u := sin(x),
dudx
= cos(x),
=∫ cos(x)
udu
cos(x)dx =
ducos(x)
=∫ 1
udu
= ln |u|+ c
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 18: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/18.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx u := sin(x),
dudx
= cos(x),
=∫ cos(x)
udu
cos(x)dx =
ducos(x)
=∫ 1
udu
= ln |u|+ c
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 19: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/19.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx u := sin(x),
dudx
= cos(x),
=∫ cos(x)
udu
cos(x)dx =
ducos(x)
=∫ 1
udu
= ln |u|+ c
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 20: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/20.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx u := sin(x),
dudx
= cos(x),
=∫ cos(x)
udu
cos(x)dx =
ducos(x)
=∫ 1
udu
= ln |u|
= ln∣∣sin(x)
∣∣µ(x) = e
∫p(x) dx = eln
∣∣sin(x)∣∣= sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 21: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/21.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx u := sin(x),
dudx
= cos(x),
=∫ cos(x)
udu
cos(x)dx =
ducos(x)
=∫ 1
udu
= ln |u|= ln
∣∣sin(x)∣∣
µ(x) = e∫
p(x) dx = eln∣∣sin(x)
∣∣= sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 22: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/22.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx u := sin(x),
dudx
= cos(x),
=∫ cos(x)
udu
cos(x)dx =
ducos(x)
=∫ 1
udu
= ln |u|= ln
∣∣sin(x)∣∣
µ(x) =
e∫
p(x) dx = eln∣∣sin(x)
∣∣= sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 23: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/23.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx u := sin(x),
dudx
= cos(x),
=∫ cos(x)
udu
cos(x)dx =
ducos(x)
=∫ 1
udu
= ln |u|= ln
∣∣sin(x)∣∣
µ(x) = e∫
p(x) dx
= eln∣∣sin(x)
∣∣= sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 24: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/24.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx u := sin(x),
dudx
= cos(x),
=∫ cos(x)
udu
cos(x)dx =
ducos(x)
=∫ 1
udu
= ln |u|= ln
∣∣sin(x)∣∣
µ(x) = e∫
p(x) dx = eln∣∣sin(x)
∣∣
= sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 25: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/25.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx u := sin(x),
dudx
= cos(x),
=∫ cos(x)
udu
cos(x)dx =
ducos(x)
=∫ 1
udu
= ln |u|= ln
∣∣sin(x)∣∣
µ(x) = e∫
p(x) dx = eln∣∣sin(x)
∣∣= sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 26: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/26.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.
y′ +cos(x)sin(x)
y = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 27: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/27.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.y′ +
cos(x)sin(x)
y = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 28: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/28.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.µ(x)
(y′ +
cos(x)sin(x)
y)
= 1 ·µ(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 29: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/29.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.sin(x)
(y′ +
cos(x)sin(x)
y)
= 1 · sin(x)
sin(x)y′ + cos(x)y = sin(x)ddx
(sin(x)y
)= sin(x)
sin(x)y = −cos(x)+ c
y = −cos(x)sin(x)
+c
sin(x)
= −cot(x)+c
sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 30: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/30.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.sin(x)
(y′ +
cos(x)sin(x)
y)
= 1 · sin(x)
sin(x)y′ + cos(x)y = sin(x)
ddx
(sin(x)y
)= sin(x)
sin(x)y = −cos(x)+ c
y = −cos(x)sin(x)
+c
sin(x)
= −cot(x)+c
sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 31: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/31.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.sin(x)
(y′ +
cos(x)sin(x)
y)
= 1 · sin(x)
sin(x)y′ + cos(x)y = sin(x)ddx
(sin(x)y
)= sin(x)
sin(x)y = −cos(x)+ c
y = −cos(x)sin(x)
+c
sin(x)
= −cot(x)+c
sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 32: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/32.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.sin(x)
(y′ +
cos(x)sin(x)
y)
= 1 · sin(x)
sin(x)y′ + cos(x)y = sin(x)ddx
(sin(x)y
)= sin(x)
sin(x)y = −cos(x)+ c
y = −cos(x)sin(x)
+c
sin(x)
= −cot(x)+c
sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 33: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/33.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.sin(x)
(y′ +
cos(x)sin(x)
y)
= 1 · sin(x)
sin(x)y′ + cos(x)y = sin(x)ddx
(sin(x)y
)= sin(x)
sin(x)y = −cos(x)+ c
y = −cos(x)sin(x)
+c
sin(x)
= −cot(x)+c
sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 34: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/34.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.sin(x)
(y′ +
cos(x)sin(x)
y)
= 1 · sin(x)
sin(x)y′ + cos(x)y = sin(x)ddx
(sin(x)y
)= sin(x)
sin(x)y = −cos(x)+ c
y = −cos(x)sin(x)
+c
sin(x)
= −cot(x)+c
sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 35: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/35.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.
y(x) = −cot(x)+c
sin(x)
0 = y(1) = −cot(1)+c
sin(1)c
sin(1)= cot(1)
c = cot(1)sin(1) = cos(1)
y(x) = −cot(x)+cos(1)sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 36: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/36.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.
y(x) = −cot(x)+c
sin(x)
0 = y(1) = −cot(1)+c
sin(1)c
sin(1)= cot(1)
c = cot(1)sin(1) = cos(1)
y(x) = −cot(x)+cos(1)sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 37: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/37.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.
y(x) = −cot(x)+c
sin(x)
0 = y(1)
= −cot(1)+c
sin(1)c
sin(1)= cot(1)
c = cot(1)sin(1) = cos(1)
y(x) = −cot(x)+cos(1)sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 38: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/38.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.
y(x) = −cot(x)+c
sin(x)
0 = y(1) = −cot(1)+c
sin(1)
csin(1)
= cot(1)
c = cot(1)sin(1) = cos(1)
y(x) = −cot(x)+cos(1)sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 39: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/39.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.
y(x) = −cot(x)+c
sin(x)
0 = y(1) = −cot(1)+c
sin(1)c
sin(1)= cot(1)
c = cot(1)sin(1) = cos(1)
y(x) = −cot(x)+cos(1)sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 40: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/40.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.
y(x) = −cot(x)+c
sin(x)
0 = y(1) = −cot(1)+c
sin(1)c
sin(1)= cot(1)
c = cot(1)sin(1)
= cos(1)
y(x) = −cot(x)+cos(1)sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 41: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/41.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.
y(x) = −cot(x)+c
sin(x)
0 = y(1) = −cot(1)+c
sin(1)c
sin(1)= cot(1)
c = cot(1)sin(1) = cos(1)
y(x) = −cot(x)+cos(1)sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 42: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/42.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.
y(x) = −cot(x)+c
sin(x)
0 = y(1) = −cot(1)+c
sin(1)c
sin(1)= cot(1)
c = cot(1)sin(1) = cos(1)
y(x) = −cot(x)+cos(1)sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 43: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/43.jpg)
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.
y(x) = −cot(x)+cos(1)sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 44: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/44.jpg)
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
y′ +cos(x)sin(x)
y ?= 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 45: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/45.jpg)
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
y′ +cos(x)sin(x)
y ?= 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 46: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/46.jpg)
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
ddx
(−cot(x)+
cos(1)sin(x)
)+
cos(x)sin(x)
(−cot(x)+
cos(1)sin(x)
)?= 1
1sin2(x)
− cos(1)cos(x)sin2(x)
− cos2(x)sin2(x)
+cos(1)cos(x)
sin2(x)?= 1
1− cos2(x)sin2(x)
?= 1
sin2(x)sin2(x)
?= 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 47: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/47.jpg)
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
ddx
(−cot(x)+
cos(1)sin(x)
)+
cos(x)sin(x)
(−cot(x)+
cos(1)sin(x)
)?= 1
1sin2(x)
− cos(1)cos(x)sin2(x)
− cos2(x)sin2(x)
+cos(1)cos(x)
sin2(x)?= 1
1− cos2(x)sin2(x)
?= 1
sin2(x)sin2(x)
?= 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 48: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/48.jpg)
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
ddx
(−cot(x)+
cos(1)sin(x)
)+
cos(x)sin(x)
(−cot(x)+
cos(1)sin(x)
)?= 1
1sin2(x)
− cos(1)cos(x)sin2(x)
− cos2(x)sin2(x)
+cos(1)cos(x)
sin2(x)?= 1
1− cos2(x)sin2(x)
?= 1
sin2(x)sin2(x)
?= 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 49: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/49.jpg)
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
ddx
(−cot(x)+
cos(1)sin(x)
)+
cos(x)sin(x)
(−cot(x)+
cos(1)sin(x)
)?= 1
1sin2(x)
− cos(1)cos(x)sin2(x)
− cos2(x)sin2(x)
+cos(1)cos(x)
sin2(x)?= 1
1− cos2(x)sin2(x)
?= 1
sin2(x)sin2(x)
?= 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 50: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/50.jpg)
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
ddx
(−cot(x)+
cos(1)sin(x)
)+
cos(x)sin(x)
(−cot(x)+
cos(1)sin(x)
)?= 1
1sin2(x)
− cos(1)cos(x)sin2(x)
− cos2(x)sin2(x)
+cos(1)cos(x)
sin2(x)
?= 1
1− cos2(x)sin2(x)
?= 1
sin2(x)sin2(x)
?= 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 51: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/51.jpg)
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
ddx
(−cot(x)+
cos(1)sin(x)
)+
cos(x)sin(x)
(−cot(x)+
cos(1)sin(x)
)?= 1
1sin2(x)
− cos(1)cos(x)sin2(x)
− cos2(x)sin2(x)
+cos(1)cos(x)
sin2(x)?= 1
1− cos2(x)sin2(x)
?= 1
sin2(x)sin2(x)
?= 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 52: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/52.jpg)
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
ddx
(−cot(x)+
cos(1)sin(x)
)+
cos(x)sin(x)
(−cot(x)+
cos(1)sin(x)
)?= 1
1sin2(x)
− cos(1)cos(x)sin2(x)
− cos2(x)sin2(x)
+cos(1)cos(x)
sin2(x)?= 1
1− cos2(x)sin2(x)
?= 1
sin2(x)sin2(x)
?= 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 53: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/53.jpg)
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
ddx
(−cot(x)+
cos(1)sin(x)
)+
cos(x)sin(x)
(−cot(x)+
cos(1)sin(x)
)?= 1
1sin2(x)
− cos(1)cos(x)sin2(x)
− cos2(x)sin2(x)
+cos(1)cos(x)
sin2(x)?= 1
1− cos2(x)sin2(x)
?= 1
sin2(x)sin2(x)
?= 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 54: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/54.jpg)
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
ddx
(−cot(x)+
cos(1)sin(x)
)+
cos(x)sin(x)
(−cot(x)+
cos(1)sin(x)
)?= 1
1sin2(x)
− cos(1)cos(x)sin2(x)
− cos2(x)sin2(x)
+cos(1)cos(x)
sin2(x)?= 1
1− cos2(x)sin2(x)
?= 1
sin2(x)sin2(x)
= 1√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 55: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/55.jpg)
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
y(1) = − cot(1)+cos(1)sin(1)
= 0√
Yes, it does
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 56: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/56.jpg)
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
y(1) =
− cot(1)+cos(1)sin(1)
= 0√
Yes, it does
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 57: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/57.jpg)
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
y(1) = − cot(1)+cos(1)sin(1)
= 0√
Yes, it does
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 58: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/58.jpg)
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
y(1) = − cot(1)+cos(1)sin(1)
= 0
√
Yes, it does
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 59: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/59.jpg)
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
y(1) = − cot(1)+cos(1)sin(1)
= 0√
Yes, it does
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
![Page 60: Linear First Order Differential Equations · 2008-11-06 · Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order](https://reader034.fdocuments.in/reader034/viewer/2022042802/5f38649bd3512815e0651f66/html5/thumbnails/60.jpg)
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
y(1) = − cot(1)+cos(1)sin(1)
= 0√
Yes, it does
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations