Help with Solving a Cauchy-Euler Differential Equation

[Jim4592]

Homework Statement

x² y'' + x y' + 4 y = 0

Homework Equations

y = x^r
y' = r x^r-1
y'' = (r²-r)x^r-2

The Attempt at a Solution

x²{(r²-r)x^r-1} + x{r x^r-1} + 4x^r

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!

Thanks in advanced!

 

[vela]

Use the definition of exponentiation: x^a=e^{a log x}.

 

[HallsofIvy]

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).