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:
- Solve the equation g(y) = 0, which gives the constant solutions of (S);
- Rewrite the equation (S) as,
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.
Example: Find the particular solution of
Solution: 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 conditiony=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