# Help with Solving a Cauchy-Euler Differential Equation

## Homework Statement

x2 y'' + x y' + 4 y = 0

y = xr
y' = r xr-1
y'' = (r2-r)xr-2

## The Attempt at a Solution

x2{(r2-r)xr-1} + x{r xr-1} + 4xr

r2 - r + r + 4

r2 + 4 = 0

r = +- 2i

y = C1x2i + c2x-2i

My question is how can i remove the imaginary number with cos[] and sin[]
If you could be as descriptive as possible i'd really appreciate it!

vela
Staff Emeritus
Homework Helper
Use the definition of exponentiation: xa=ea log x.

HallsofIvy
Homework Helper
In fact, the substitution y= ln(x) will change any "Cauchy-Euler" equation into an equation with constant coefficients with the same characteristic equation. An equation with constant coefficients, with $\pm 2$ as characteristic roots, has general solution C cos(2y)+ D sin(2y).