Separation of variables is a method to solve differential equations.
A DE is call separable if you can use isolate the two variables on opposite sides of the equation.
Example: solve
can be rewritten as , and we can integrate both side We can amalgamate the two constants of integration into one constant:
Next we solve for
as a function of . The absolute value signs can be removed, but then the right hand side might be positive or negative. We write this as
Finally we replace the constant
by to get the solution
Lost Solutions
When applying the separation of variables technique, it’s crucial to be aware that some solutions may be inadvertently excluded from the final result. One reason for that is assumptions made during the separation process.
has a trivial solution , but separation of variables will not find this solution Let’s solve the equation by separation of variables
- Separate variables:
. - Integrate:
. - Solve for
: . The constant solution get lost at step 1, where the form is only valid when .
In general, for a separable DE