SEPARATION OF VARIABLES
Differential Equations
SEPARATION OF VARIABLES
Graham S McDonald A Tutorial Module for learning the technique
of separation of variables
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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.
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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
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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
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