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.

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

  1. Separate variables: .
  2. Integrate: .
  3. Solve for : . The constant solution get lost at step 1, where the form is only valid when .

In general, for a separable DE , all the roots of gives lost solutions.