# Separable Equations

The differential equation of the
form is called
**separable**, if *f*(*x*,*y*) =
*h*(*x*) *g*(*y*); that is,

In
order to solve it, perform the following steps:

**(1)** - Solve the equation
*g*(*y*) = 0, which gives the constant solutions of
(*S*); **(2)** - Rewrite the equation
(
*S*) as ,

and, then, integrate

to obtain

**(3)** - Write down all the solutions; the
constant ones obtained from (1) and the ones given in (2);
**(4)** - If you are given an IVP, use the initial
condition to find the particular solution. Note that it may happen
that the particular solution is one of the constant solutions given in
(1). This is why Step 3 is important.

**Example:**
Find the particular solution of

**Solution:**
Perform the following steps:

**(1)** -
In order to find the constant solutions, solve .
We obtain
*y* = 1 and *y*=-1.
**(2)**
- Rewrite the equation as
.

Using the techniques of
integration of rational functions, we get

,

which implies

**(3)**
- The solutions to the given differential equation are

**(4)**
- Since the constant solutions do not satisfy the initial
condition, we are left to find the particular solution among the ones
found in (2), that is we need to find the constant
*C*. If we plug in
the condition *y*=2 when *x*=1, we get
.

Note that this solution is given in an implicit form. You may be asked
to rewrite it in an explicit one. For example, in this case, we have

### If you would like more practice, click on
Example.

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*
Author: Mohamed
Amine Khamsi *

Last Update 6-22-98

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