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:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \frac{d}{dx} (P(x) \frac{du}{dx}) + Q(x)u = 0 [/tex]

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Existence of solution in Sturm-Liouville DE

Loading...

**Physics Forums | Science Articles, Homework Help, Discussion**