Differential Equations
EXACT EQUATIONS
Graham S McDonald
A Tutorial Module for learning the technique
of solving exact differential equations
● Table of contents
● Begin Tutorial
c 2004
, 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
∂P ∂Q
If ∂y = ∂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 .
∂u ∂u
One solves ∂x = P and ∂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.
Toc JJ II J I Back
, 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 − 2 dx = 0
x x
Exercise 2.
dy
2xy + y 2 − 2x = 0
dx
Exercise 3.
2(y + 1)ex dx + 2(ex − 2y)dy = 0
● Theory ● Answers ● Integrals ● Tips
Toc JJ II J I Back
EXACT EQUATIONS
Graham S McDonald
A Tutorial Module for learning the technique
of solving exact differential equations
● Table of contents
● Begin Tutorial
c 2004
, 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
∂P ∂Q
If ∂y = ∂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 .
∂u ∂u
One solves ∂x = P and ∂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.
Toc JJ II J I Back
, 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 − 2 dx = 0
x x
Exercise 2.
dy
2xy + y 2 − 2x = 0
dx
Exercise 3.
2(y + 1)ex dx + 2(ex − 2y)dy = 0
● Theory ● Answers ● Integrals ● Tips
Toc JJ II J I Back
