Differential Equations EXACT EQUATIONS

[Pages:34]Differential Equations

EXACT EQUATIONS

Graham S McDonald A Tutorial Module for learning the technique

of solving exact 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. Standard integrals 5. Tips on using solutions

Full worked solutions

Section 1: Theory

3

1. Theory

We consider here the following standard form of ordinary differential equation (o.d.e.):

P (x, y)dx + Q(x, y)dy = 0

If

P y

=

Q x

then

the

o.de.

is

said

to

be

exact.

This means that a function u(x, y) exists such that:

u

u

du = dx + dy

x

y

= P dx + Q dy = 0 .

One

solves

u x

=

P

and

u y

=

Q

to

find

u(x, y).

Then du = 0 gives u(x, y) = C, where C is a constant. This last equation gives the general solution of P dx + Q dy = 0.

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Section 2: Exercises

4

2. Exercises

Click on Exercise links for full worked solutions (there are 11 exercises in total)

Show that each of the following differential equations is exact and use that property to find the general solution:

Exercise 1.

1

y

dy x

-

x2

dx

=

0

Exercise 2.

dy 2xy

+

y2

-

2x

=

0

dx

Exercise 3. 2(y + 1)exdx + 2(ex - 2y)dy = 0

q Theory q Answers q Integrals q Tips

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Section 2: Exercises

5

Exercise 4. (2xy + 6x)dx + (x2 + 4y3)dy = 0

Exercise 5. (8y - x2y) dy + x - xy2 = 0

dx

Exercise 6. (e4x + 2xy2)dx + (cos y + 2x2y)dy = 0

Exercise 7. (3x2 + y cos x)dx + (sin x - 4y3)dy = 0

q Theory q Answers q Integrals q Tips

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Section 2: Exercises

6

Exercise 8.

x

tan-1

y

?

dx

+

x2 2(1 +

y2)

?

dy

=

0

Exercise 9. (2x + x2y3)dx + (x3y2 + 4y3)dy = 0

Exercise 10. (2x3 - 3x2y + y3) dy = 2x3 - 6x2y + 3xy2

dx

Exercise 11. (y2 cos x - sin x)dx + (2y sin x + 2)dy = 0

q Theory q Answers q Integrals q Tips

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Section 3: Answers

3. Answers

1. y = Ax ,

2. y2x - x2 = A , 3. (y + 1)ex - y2 = A , 4. x2y + 3x2 + y4 = A ,

5.

1 2

x2(1

-

y2)

+

4y2

=

A

,

6.

1 4

e4x

+

x2y2

+

sin

y

=

A

,

7. x3 + y sin x - y4 = A ,

8.

x2 2

tan-1

y

=

A

,

9.

x2

+

x3 y 3 3

+ y4 = A ,

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Section 3: Answers

8

10.

x4 2

- 2x3y +

3 2

x2y2

-

y4 4

=A,

11. y2 sin x + cos x + 2y = A.

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