Differential Equations BERNOULLI EQUATIONS

[Pages:31]Differential Equations

BERNOULLI EQUATIONS

Graham S McDonald A Tutorial Module for learning how to solve

Bernoulli differential equations

q Table of contents q Begin Tutorial

c 2004 g.s.mcdonald@salford.ac.uk

Table of contents

1. Theory 2. Exercises 3. Answers 4. Integrating factor method 5. Standard integrals 6. Tips on using solutions

Full worked solutions

Section 1: Theory

3

1. Theory

A Bernoulli differential equation can be written in the following

standard form:

dy + P (x)y = Q(x)yn , dx

where n = 1 (the equation is thus nonlinear).

To find the solution, change the dependent variable from y to z, where z = y1-n. This gives a differential equation in x and z that is

linear, and can be solved using the integrating factor method.

Note: Dividing the above standard form by yn gives:

1 yn

dy dx

+

P (x)y1-n

=

Q(x)

1 dz

i.e.

+ P (x)z = Q(x)

(1 - n) dx

where

we

have

used

dz dx

=

(1

-

n)y-n

dy dx

.

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Section 2: Exercises

4

2. Exercises

Click on Exercise links for full worked solutions (there are 9 exercises in total)

Exercise 1. The general form of a Bernoulli equation is

dy + P (x)y = Q(x) yn , dx where P and Q are functions of x, and n is a constant. Show that the transformation to a new dependent variable z = y1-n reduces the equation to one that is linear in z (and hence solvable using the integrating factor method).

Solve the following Bernoulli differential equations:

Exercise 2.

dy

-

1 y

=

xy2

dx x

q Theory q Answers q IF method q Integrals q Tips

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Section 2: Exercises

5

Exercise 3. dy + y = y2 dx x

Exercise 4.

dy

+

1 y

=

exy4

dx 3

Exercise 5.

dy x

+y

=

xy3

dx

Exercise 6.

dy

+

2 y

=

-x2

cos x ?

y2

dx x

q Theory q Answers q IF method q Integrals q Tips

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Section 2: Exercises

6

Exercise 7.

dy 2

+ tan x ? y

=

(4x +

5)2 y3

dx

cos x

Exercise 8.

dy x

+y

=

y2x2 ln

x

dx

Exercise 9. dy = y cot x + y3cosecx dx

q Theory q Answers q IF method q Integrals q Tips

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Section 3: Answers

7

3. Answers

1. HINT: Firstly, divide each term by yn. Then, differentiate z

with

respect

to

x

to

show

that

1 dz (1-n) dx

=

1 yn

dy dx

,

2.

1 y

=

-

x2 3

+

C x

,

3.

1 y

=

x(C

-

ln

x)

,

4.

1 y3

= ex(C - 3x) ,

5.

y2 =

1 2x+C x2

,

6.

1 y

=

x2(sin x

+ C) ,

7.

1 y2

=

12

1 cos

x (4x

+

5)3

+

C cos x

,

8.

1 xy

=

C

+ x(1 -

ln

x) ,

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Section 3: Answers

8

9.

y2 =

sin2 x 2 cos x+C

.

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