[PPT] - Bernoulli Equations Bernd Schr oder logo1 Bernd Schr oder PowerPoint Presentation

SLIDE 1

logo1 Overview An Example Double Check

Bernoulli Equations

Bernd Schr¨

der

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 2

logo1 Overview An Example Double Check

What are Bernoulli Equations?

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 3

logo1 Overview An Example Double Check

What are Bernoulli Equations?

1. A Bernoulli equation is of the form y′ +p(x)y = q(x)yn,

where n = 0,1.

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 4

logo1 Overview An Example Double Check

What are Bernoulli Equations?

1. A Bernoulli equation is of the form y′ +p(x)y = q(x)yn,

where n = 0,1.

2. Recognizing Bernoulli equations requires some pattern

recognition.

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 5

logo1 Overview An Example Double Check

What are Bernoulli Equations?

1. A Bernoulli equation is of the form y′ +p(x)y = q(x)yn,

where n = 0,1.

2. Recognizing Bernoulli equations requires some pattern

recognition.

3. To solve a Bernoulli equation, we translate the equation

into a linear equation.

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 6

logo1 Overview An Example Double Check

What are Bernoulli Equations?

1. A Bernoulli equation is of the form y′ +p(x)y = q(x)yn,

where n = 0,1.

2. Recognizing Bernoulli equations requires some pattern

recognition.

3. To solve a Bernoulli equation, we translate the equation

into a linear equation.

3.1 The substitution y = v

1 1−n turns the Bernoulli equation

y′ +p(x)y = q(x)yn into a linear first order equation for v,

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 7

logo1 Overview An Example Double Check

What are Bernoulli Equations?

1. A Bernoulli equation is of the form y′ +p(x)y = q(x)yn,

where n = 0,1.

2. Recognizing Bernoulli equations requires some pattern

recognition.

3. To solve a Bernoulli equation, we translate the equation

into a linear equation.

3.1 The substitution y = v

1 1−n turns the Bernoulli equation

y′ +p(x)y = q(x)yn into a linear first order equation for v, 3.2 We can even write down the abstract form of the resulting linear first order equation, but it is simpler to remember the substitution y = v

1 1−n , Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 8

logo1 Overview An Example Double Check

What are Bernoulli Equations?

1. A Bernoulli equation is of the form y′ +p(x)y = q(x)yn,

where n = 0,1.

2. Recognizing Bernoulli equations requires some pattern

recognition.

3. To solve a Bernoulli equation, we translate the equation

into a linear equation.

3.1 The substitution y = v

1 1−n turns the Bernoulli equation

y′ +p(x)y = q(x)yn into a linear first order equation for v, 3.2 We can even write down the abstract form of the resulting linear first order equation, but it is simpler to remember the substitution y = v

1 1−n ,

3.3 After we solve the equation for v, we obtain y as the appropriate power of v.

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 9

logo1 Overview An Example Double Check

What are Bernoulli Equations?

1. A Bernoulli equation is of the form y′ +p(x)y = q(x)yn,

where n = 0,1.

2. Recognizing Bernoulli equations requires some pattern

recognition.

3. To solve a Bernoulli equation, we translate the equation

into a linear equation.

3.1 The substitution y = v

1 1−n turns the Bernoulli equation

y′ +p(x)y = q(x)yn into a linear first order equation for v, 3.2 We can even write down the abstract form of the resulting linear first order equation, but it is simpler to remember the substitution y = v

1 1−n ,

3.3 After we solve the equation for v, we obtain y as the appropriate power of v.

That’s it.

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 10

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 11

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Reduction to a linear equation.

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 12

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Reduction to a linear equation. n = 5,

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 13

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Reduction to a linear equation. n = 5, y = v

1 1−5

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 14

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Reduction to a linear equation. n = 5, y = v

1 1−5 = v− 1 4,

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 15

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Reduction to a linear equation. n = 5, y = v

1 1−5 = v− 1 4,

y′ =

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 16

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Reduction to a linear equation. n = 5, y = v

1 1−5 = v− 1 4,

y′ = d dxv− 1

4

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 17

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Reduction to a linear equation. n = 5, y = v

1 1−5 = v− 1 4,

y′ = d dxv− 1

4

= −1 4v− 5

4v′

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 18

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Reduction to a linear equation. n = 5, y = v

1 1−5 = v− 1 4,

y′ = d dxv− 1

4

= −1 4v− 5

4v′

y′ +y = x2y5

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 19

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Reduction to a linear equation. n = 5, y = v

1 1−5 = v− 1 4,

y′ = d dxv− 1

4

= −1 4v− 5

4v′

y′ +y = x2 v− 1

4

5

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 20

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Reduction to a linear equation. n = 5, y = v

1 1−5 = v− 1 4,

y′ = d dxv− 1

4

= −1 4v− 5

4v′

y′ +

v− 1

4

=

x2 v− 1

4

5

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 21

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Reduction to a linear equation. n = 5, y = v

1 1−5 = v− 1 4,

y′ = d dxv− 1

4

= −1 4v− 5

4v′

−1 4v− 5

4v′ +

v− 1

4

=

x2 v− 1

4

5

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 22

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Reduction to a linear equation. n = 5, y = v

1 1−5 = v− 1 4,

y′ = d dxv− 1

4

= −1 4v− 5

4v′

−1 4v− 5

4v′ +

v− 1

4

=

x2 v− 1

4

5 −1 4v− 5

4v′ +v− 1 4

= x2v− 5

4

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 23

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Reduction to a linear equation. n = 5, y = v

1 1−5 = v− 1 4,

y′ = d dxv− 1

4

= −1 4v− 5

4v′

−1 4v− 5

4v′ +

v− 1

4

=

x2 v− 1

4

5 −1 4v− 5

4v′ +v− 1 4

= x2v− 5

4

−1 4v′ +v = x2

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 24

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Reduction to a linear equation. n = 5, y = v

1 1−5 = v− 1 4,

y′ = d dxv− 1

4

= −1 4v− 5

4v′

−1 4v− 5

4v′ +

v− 1

4

=

x2 v− 1

4

5 −1 4v− 5

4v′ +v− 1 4

= x2v− 5

4

−1 4v′ +v = x2 v′ −4v = −4x2

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 25

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Solving the linear equation.

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 26

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Solving the linear equation. v′ −4v = −4x2

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 27

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Solving the linear equation. v′ −4v = −4x2 µ(x)

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 28

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Solving the linear equation. v′ −4v = −4x2 µ(x) = e

p(x) dx

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 29

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Solving the linear equation. v′ −4v = −4x2 µ(x) = e

p(x) dx = e −4 dx

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 30

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Solving the linear equation. v′ −4v = −4x2 µ(x) = e

p(x) dx = e −4 dx

= e−4x

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 31

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Solving the linear equation. v′ −4v = −4x2 µ(x) = e

p(x) dx = e −4 dx

= e−4x e−4x v′ −4v

=

−4x2e−4x

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 32

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Solving the linear equation. v′ −4v = −4x2 µ(x) = e

p(x) dx = e −4 dx

= e−4x e−4x v′ −4v

=

−4x2e−4x e−4xv′ −4e−4xv = −4x2e−4x

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 33

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Solving the linear equation. v′ −4v = −4x2 µ(x) = e

p(x) dx = e −4 dx

= e−4x e−4x v′ −4v

=

−4x2e−4x e−4xv′ −4e−4xv = −4x2e−4x

e−4xv

′ = −4x2e−4x

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 34

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Solving the linear equation. v′ −4v = −4x2 µ(x) = e

p(x) dx = e −4 dx

= e−4x e−4x v′ −4v

=

−4x2e−4x e−4xv′ −4e−4xv = −4x2e−4x

e−4xv

′ = −4x2e−4x e−4xv =

−4x2e−4x dx

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 35

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Solving the linear equation. v′ −4v = −4x2 µ(x) = e

p(x) dx = e −4 dx

= e−4x e−4x v′ −4v

=

−4x2e−4x e−4xv′ −4e−4xv = −4x2e−4x

e−4xv

′ = −4x2e−4x e−4xv =

−4x2e−4x dx

v = −4e4x

x2e−4x dx

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 36

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Solving the linear equation (continued).

x2e−4x dx

=

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 37

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Solving the linear equation (continued).

x2e−4x dx

= − 1 4e−4xx2 − −1 4

e−4x2x dx

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 38

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Solving the linear equation (continued).

x2e−4x dx

= − 1 4e−4xx2 − −1 4

e−4x2x dx

= −1 4e−4xx2 + 1 2

e−4xx dx

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 39

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Solving the linear equation (continued).

x2e−4x dx

= − 1 4e−4xx2 − −1 4

e−4x2x dx

= −1 4e−4xx2 + 1 2

e−4xx dx

= −1 4e−4xx2 + 1 2

−1

4e−4xx−

−1

4e−4x dx

Bernd Schr¨
der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 40

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Solving the linear equation (continued).

x2e−4x dx

= − 1 4e−4xx2 − −1 4

e−4x2x dx

= −1 4e−4xx2 + 1 2

e−4xx dx

= −1 4e−4xx2 + 1 2

−1

4e−4xx−

−1

4e−4x dx

=

−1 4e−4xx2 − 1 8e−4xx+ 1 8

e−4x dx

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 41

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Solving the linear equation (continued).

x2e−4x dx

= − 1 4e−4xx2 − −1 4

e−4x2x dx

= −1 4e−4xx2 + 1 2

e−4xx dx

= −1 4e−4xx2 + 1 2

−1

4e−4xx−

−1

4e−4x dx

=

−1 4e−4xx2 − 1 8e−4xx+ 1 8

e−4x dx

= −1 4e−4xx2 − 1 8e−4xx− 1 32e−4x +c

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 42

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Solving the linear equation (concluded), general solution of the

riginal equation.

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 43

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Solving the linear equation (concluded), general solution of the

riginal equation.

v = −4e4x

−1

4e−4xx2 − 1 8e−4xx− 1 32e−4x +c

Bernd Schr¨
der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 44

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Solving the linear equation (concluded), general solution of the

riginal equation.

v = −4e4x

−1

4e−4xx2 − 1 8e−4xx− 1 32e−4x +c

=

x2 + 1 2x+ 1 8 +ce4x

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 45

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Solving the linear equation (concluded), general solution of the

riginal equation.

v = −4e4x

−1

4e−4xx2 − 1 8e−4xx− 1 32e−4x +c

=

x2 + 1 2x+ 1 8 +ce4x y = v− 1

4

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 46

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Solving the linear equation (concluded), general solution of the

riginal equation.

v = −4e4x

−1

4e−4xx2 − 1 8e−4xx− 1 32e−4x +c

=

x2 + 1 2x+ 1 8 +ce4x y = v− 1

4 =

x2 + 1

2x+ 1 8 +ce4x − 1

4

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 47

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Finding c.

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 48

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Finding c. y =

x2 + 1

2x+ 1 8 +ce4x − 1

4

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 49

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Finding c. y =

x2 + 1

2x+ 1 8 +ce4x − 1

4

1 = y(0)

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 50

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Finding c. y =

x2 + 1

2x+ 1 8 +ce4x − 1

4

1 = y(0) =

02 + 1

2 ·0+ 1 8 +ce4·0 − 1

4

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 51

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Finding c. y =

x2 + 1

2x+ 1 8 +ce4x − 1

4

1 = y(0) =

02 + 1

2 ·0+ 1 8 +ce4·0 − 1

4

1 = 1 8 +c − 1

4

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 52

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Finding c. y =

x2 + 1

2x+ 1 8 +ce4x − 1

4

1 = y(0) =

02 + 1

2 ·0+ 1 8 +ce4·0 − 1

4

1 = 1 8 +c − 1

4

1 = 1 8 +c

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 53

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Finding c. y =

x2 + 1

2x+ 1 8 +ce4x − 1

4

1 = y(0) =

02 + 1

2 ·0+ 1 8 +ce4·0 − 1

4

1 = 1 8 +c − 1

4

1 = 1 8 +c c = 7 8,

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 54

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

Finding c. y =

x2 + 1

2x+ 1 8 +ce4x − 1

4

1 = y(0) =

02 + 1

2 ·0+ 1 8 +ce4·0 − 1

4

1 = 1 8 +c − 1

4

1 = 1 8 +c c = 7 8, y =

x2 + 1

2x+ 1 8 + 7 8e4x − 1

4

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 55

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1.

y =

x2 + 1

2x+ 1 8 + 7 8e4x − 1

4

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 56

logo1 Overview An Example Double Check

Does y =

x2 + 1

2x+ 1 8 + 7 8e4x −1

4

Really Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1?

d dx

x2 + 1

2x+ 1 8 + 7 8e4x − 1

4

+
x2 + 1

2x+ 1 8 + 7 8e4x − 1

4 Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 57

logo1 Overview An Example Double Check

Does y =

x2 + 1

2x+ 1 8 + 7 8e4x −1

4

Really Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1?

d dx

x2 + 1

2x+ 1 8 + 7 8e4x − 1

4

+
x2 + 1

2x+ 1 8 + 7 8e4x − 1

4

= −1 4

x2 + 1

2x+ 1 8 + 7 8e4x − 5

4

2x+ 1 2 + 7 2e4x

+
x2 + 1

2x+ 1 8 + 7 8e4x − 1

4 Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 58

logo1 Overview An Example Double Check

Does y =

x2 + 1

2x+ 1 8 + 7 8e4x −1

4

Really Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1?

d dx

x2 + 1

2x+ 1 8 + 7 8e4x − 1

4

+
x2 + 1

2x+ 1 8 + 7 8e4x − 1

4

= −1 4

x2 + 1

2x+ 1 8 + 7 8e4x − 5

4

2x+ 1 2 + 7 2e4x

+
x2 + 1

2x+ 1 8 + 7 8e4x − 1

4

=

x2 + 1

2x+ 1 8 + 7 8e4x − 5

4

−x 2 − 1 8 − 7 8e4x +x2 + 1 2x+ 1 8 + 7 8e4x

Bernd Schr¨
der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 59

logo1 Overview An Example Double Check

Does y =

x2 + 1

2x+ 1 8 + 7 8e4x −1

4

Really Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1?

d dx

x2 + 1

2x+ 1 8 + 7 8e4x − 1

4

+
x2 + 1

2x+ 1 8 + 7 8e4x − 1

4

= −1 4

x2 + 1

2x+ 1 8 + 7 8e4x − 5

4

2x+ 1 2 + 7 2e4x

+
x2 + 1

2x+ 1 8 + 7 8e4x − 1

4

=

x2 + 1

2x+ 1 8 + 7 8e4x − 5

4

−x 2 − 1 8 − 7 8e4x +x2 + 1 2x+ 1 8 + 7 8e4x

=
x2 + 1

2x+ 1 8 + 7 8e4x − 5

4

x2

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 60

logo1 Overview An Example Double Check

Does y =

x2 + 1

2x+ 1 8 + 7 8e4x −1

4

Really Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1?

d dx

x2 + 1

2x+ 1 8 + 7 8e4x − 1

4

+
x2 + 1

2x+ 1 8 + 7 8e4x − 1

4

= −1 4

x2 + 1

2x+ 1 8 + 7 8e4x − 5

4

2x+ 1 2 + 7 2e4x

+
x2 + 1

2x+ 1 8 + 7 8e4x − 1

4

=

x2 + 1

2x+ 1 8 + 7 8e4x − 5

4

−x 2 − 1 8 − 7 8e4x +x2 + 1 2x+ 1 8 + 7 8e4x

=
x2 + 1

2x+ 1 8 + 7 8e4x − 5

4

x2 = y5x2

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 61

logo1 Overview An Example Double Check

Does y =

x2 + 1

2x+ 1 8 + 7 8e4x −1

4

Really Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1?

d dx

x2 + 1

2x+ 1 8 + 7 8e4x − 1

4

+
x2 + 1

2x+ 1 8 + 7 8e4x − 1

4

= −1 4

x2 + 1

2x+ 1 8 + 7 8e4x − 5

4

2x+ 1 2 + 7 2e4x

+
x2 + 1

2x+ 1 8 + 7 8e4x − 1

4

=

x2 + 1

2x+ 1 8 + 7 8e4x − 5

4

−x 2 − 1 8 − 7 8e4x +x2 + 1 2x+ 1 8 + 7 8e4x

=
x2 + 1

2x+ 1 8 + 7 8e4x − 5

4

x2 = y5x2 √

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 62

logo1 Overview An Example Double Check

Does y =

x2 + 1

2x+ 1 8 + 7 8e4x −1

4

Really Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1?

y(0) =

02 + 1

2 ·0+ 1 8 + 7 8e4·0 − 1

4

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 63

logo1 Overview An Example Double Check

Does y =

x2 + 1

2x+ 1 8 + 7 8e4x −1

4

Really Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1?

y(0) =

02 + 1

2 ·0+ 1 8 + 7 8e4·0 − 1

4

= 1 8 + 7 8 − 1

4

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 64

logo1 Overview An Example Double Check

Does y =

x2 + 1

2x+ 1 8 + 7 8e4x −1

4

Really Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1?

y(0) =

02 + 1

2 ·0+ 1 8 + 7 8e4·0 − 1

4

= 1 8 + 7 8 − 1

4

= 1

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 65

logo1 Overview An Example Double Check

Does y =

x2 + 1

2x+ 1 8 + 7 8e4x −1

4

Really Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1?

y(0) =

02 + 1

2 ·0+ 1 8 + 7 8e4·0 − 1

4

= 1 8 + 7 8 − 1

4

= 1 √

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations

SLIDE 66

logo1 Overview An Example Double Check

Does y =

x2 + 1

2x+ 1 8 + 7 8e4x −1

4

Really Solve the Initial Value Problem y′ +y = x2y5, y(0) = 1?

y(0) =

02 + 1

2 ·0+ 1 8 + 7 8e4·0 − 1

4

= 1 8 + 7 8 − 1

4

= 1 √ Yes, it does.

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Bernoulli Equations