# Existence of solution in Sturm-Liouville DE

1. Mar 2, 2005

### meteorologist1

Hi, I have trouble constructing the proof for the existence of a solution u that vanishes at some point in an open interval (a, b) for the Sturm-Liouville differential equation:

$$\frac{d}{dx} (P(x) \frac{du}{dx}) + Q(x)u = 0$$

We can assume that P is continuously differentiable and greater than 0 in the closed interval [a, b], and Q is continuous on [a, b].

I don't know if it's true that for any second-order ODE, there exists a basis of solutions u1 and u2. Does anyone know? If so, since the Sturm-Liouville DE is of second-order, let u1 and u2 be a basis of solutions, and pick a point c in (a, b). Then we have u(c) = k u1(c) + m u2(c) = 0 by choosing appropriate constants k and m. Not sure if this is right or not.

Thanks.

2. Mar 4, 2005

### Feynman

We can use the weak variational problem then Lax-Miligram theorem