SEPARATION OF VARIABLES

[Pages:37]Differential Equations

SEPARATION OF VARIABLES

Graham S McDonald A Tutorial Module for learning the technique

of separation of variables

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. Standard integrals 5. Tips on using solutions

Full worked solutions

Section 1: Theory

3

1. Theory

If one can re-arrange an ordinary differential equation into the following standard form:

dy = f (x)g(y),

dx

then the solution may be found by the technique of SEPARATION OF VARIABLES:

dy = f (x) dx .

g(y)

This result is obtained by dividing the standard form by g(y), and then integrating both sides with respect to x.

Toc

Back

Section 2: Exercises

4

2. Exercises

Click on Exercise links for full worked solutions (there are 16 exercises in total)

Exercise 1. Find the general solution of dy = 3x2e-y and the particular solution

dx that satisfies the condition y(0) = 1

Exercise 2. dy y

Find the general solution of = dx x

Exercise 3.

dy y + 1

Solve the equation =

given the boundary condition: y = 1

dx x - 1

at x = 0

q Theory q Answers q Integrals q Tips

Toc

Back

Section 2: Exercises

5

Exercise 4. Solve y2 dy = x and find the particular solution when y(0) = 1

dx

Exercise 5. Find the solution of dy = e2x+y that has y = 0 when x = 0

dx

Exercise 6.

xy dy

Find the general solution of

=

x + 1 dx

Exercise 7. Find the general solution of x sin2 y. dy = (x + 1)2

dx

q Theory q Answers q Integrals q Tips

Toc

Back

Section 2: Exercises

6

Exercise 8.

Solve

dy dx

= -2x tan y

subject

to

the

condition:

y

=

2

when

x=0

Exercise 9. Solve (1 + x2) dy + xy = 0

dx and find the particular solution when

y(0) = 2

Exercise 10.

dy Solve x

= y2 + 1

and find the particular solution when

y(1) = 1

dx

Exercise 11. Find the general solution of x dy = y2 - 1

dx

q Theory q Answers q Integrals q Tips

Toc

Back

Section 2: Exercises

7

Exercise 12.

1 dy

x

Find the general solution of y dx = x2 + 1

Exercise 13.

dy

y

Solve =

and find the particular solution when y(1) = 3

dx x(x + 1)

Exercise 14. Find the general solution of sec x ? dy = sec2 y

dx

Exercise 15. Find the general solution of cosec3x dy = cos2 y

dx

q Theory q Answers q Integrals q Tips

Toc

Back

Section 2: Exercises

8

Exercise 16. Find the general solution of (1 - x2) dy + x(y - a) = 0 , where a is

dx a constant

q Theory q Answers q Integrals q Tips

Toc

Back

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download