Help finding a general solution for an eigenvalue problem

[YellowJacket]

Homework Statement

Hey, guys. I'm having trouble finding the general solution to a second order, homogeneous ODE. It is the first step to solving an eigenvalue problem and my professor is about as much help as a hole in the head. I've tried multiple "guesses" and have combed various resources looking for general solutions and have come up empty. Can someone help out, please?

Homework Equations

X''(x) + X'(x) - λX(x) = 0
X(0) = 0, X(a) = 0

The Attempt at a Solution

I cannot even solve the characteristic equation, m^2 + m - λ = 0 because you end up getting √1+4λ under the radical in the quadratic equation. In summary, I'm stuck, haha.

 

[HallsofIvy]

What's wrong with that? $\lambda$ is just a number. There is no reason in the world why it could not be "under the radical".

By the way, it is not $\sqrt{1- \lambda^2}$. By the quadratic formula, the solution to $m^2+ m- \lambda= 0$ is
$$\frac{-1\pm\sqrt{1^2- 4(1)(-\lambda)}}{2(1)}= \frac{-1\pm\sqrt{1+ 4\lambda}}{2}$$

Now, you would want to consider the cases $1+ 4\lambda> 0$, $1+ 4\lambda= 0$, and $1+ 4\lambda< 0$ separately.