# Why does this blow up?

1. ### rsq_a

108
I'm looking at the BVP:

$$y'' + ay' + e^{ax}y = 1$$,

with y(0) = 0 and y(10) = 0.

The numerical solution blows up at certain values of $$a$$. For example, a near 0.089 and a near 0.2302. Why does this happen and how do I predict it?

2. ### rsq_a

108
Erm. I found the problem. Near those values of 'a', there exists a zero eigenvalue of the linear operator. I guess that means that,

$$y'' + ay' + e^{ax}y = 0\cdot u^* = 1$$,

is a possible solution, and thus the eigenfunction $$u^* \to \infty$$ will cause the blowup.

Is this correct? It's been a while since I've done Sturm-Liouville stuff.

22
What is x?

4. ### rsq_a

108
$$y=y(x)$$