Euler s ODE

The differential equations variant coefficients
A second order nonhomogeneous equation with variant coefficients is written as

I- Euler s ODE
In this section we want to look for solutions to the differential equation in the form

These type of differential equations are called Euler equations.
This differential equation can be transformed in to a second order homogeneous equation with constant coefficients with use of the substitution:
Differentiating with respect to gives:

In the Euler equation we see the term which we need to substitute for, so we will multiply both sides of


This simplifies to
Differentiating this with respect to gives:
In the Euler equation we see the term which we need to substitute for, so we will multiply the left side of the above equation by and the right hand side by (Remember they are equal) gives:

So under this substitution the Euler equation becomes:

This is a second order linear equation with constant coefficients.
Example 1:
Solution : The given differential equation is Euler equation
Put









Example 2:
Solution : The given differential equation is Euler equation
Put









Example 3:
Solution : The given differential equation is Euler equation
Put












Solve the differential equations