The differential equation of the
form is called
separable, if f(x,y) =
h(x) g(y); that is,
order to solve it, perform the following steps:
- Solve the equation
g(y) = 0, which gives the constant solutions of
- Rewrite the equation
and, then, integrate
- Write down all the solutions; the
constant ones obtained from (1) and the ones given in (2);
- 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.
Find the particular solution of
Perform the following steps:
In order to find the constant solutions, solve .
We obtain y = 1 and y=-1.
- Rewrite the equation as
Using the techniques of
integration of rational functions, we get
- The solutions to the given differential equation are
- 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
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