Euler Equations

In the case that the functions , and of a differential equation

are not constants then it is much harder to solve it. In this section we consider the special case when , , and where and are constants. This equation is called Euler equation. First we solve the homogeneous equation , and then we use The Method of Variations to find a particular solution . Solutions of the homogeneous equation depend on the roots of the characteristic equation . If the roots and are real and different then the solution is . If the roots and are real and equal then the solution is . If the roots and are complex with and being the real and imaginary part then the solution is

. We ilustrate this on the following examples.

Example 1: Solve .

Example 2: Solve .