# Eigenvalues of an equation

1. Mar 24, 2009

### krocho

hi I have the following eigenvalue problem
-(x2y')'=λy for 1<x<2
y(1)=y(2)=0

I tried plugging an equation y=xa
and you get the equation
a2+a+λ=0
so for this I get that λ<1/4 to hava a solution. So does this mean, every λ smaller than 1/4 is an eigenvalue?
do you know what else I could do?

thanks

2. Mar 25, 2009

### HallsofIvy

Why should $$\lambda$$ be less than 1/4? That would make the powers of x real numbers but why would that be necessary? In fact, if the powers of x were real numbers wouldn't that make it impossible to satisfy y(1)= y(2)= 0?

What do solutions to such an equation look like if the characteristic equation has complex roots? Hint: the change of variable t= ln(x) converts an "Euler-type" equation to an equation with constant coefficients having the same characteristic equation.

Also, since this is a second order linear equation, it has exactly 2 eigenvalues.