Existence of solution in Sturm-Liouville DE

[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.

 

[Feynman]

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

 

[M98Ranger]

Hello! Thank you for sharing your question about the existence of solutions in the Sturm-Liouville differential equation. Let me try to provide some insight and guidance on constructing a proof for the existence of a solution that vanishes at a point in the open interval (a, b).

First, let's start by defining what it means for a solution u to vanish at a point c in the interval (a, b). This means that u(c) = 0, or in other words, the function u has a root at c.

Now, for any second-order ODE, it is true that there exists a basis of solutions u1 and u2. This is known as the fundamental set of solutions, and it is a result of the existence and uniqueness theorem for second-order linear differential equations.

With this in mind, let's consider the Sturm-Liouville DE. We have the equation:

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

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

Now, let u1 and u2 be a basis of solutions for this DE. We can then write the general solution to this DE as:

u(x) = k u1(x) + m u2(x)

where k and m are constants.

Next, we want to show that there exists a solution u that vanishes at a point c in the interval (a, b). This means that u(c) = 0, or in other words, we want to find values of k and m such that:

u(c) = k u1(c) + m u2(c) = 0

Since u1 and u2 are solutions to the DE, we know that they satisfy the equation. Therefore, we can substitute them into our equation and solve for k and m:

\frac{d}{dx} (P(x) \frac{du1}{dx}) + Q(x)u1 = 0

\frac{d}{dx} (P(x) \frac{du2}{dx}) + Q(x)u2 = 0

Substituting these into the equation u(c) = k u1(c) + m u2(c) = 0, we get:

\frac{d}{dx}