Prove that $y_1=y_2$ or $\{x \in [a,b] \mid y_1(x)=y_2(x)\}$ is finite

Question

Let $y_1$ and $y_2$ be solutions of $y''+P(x)y'+Q(x)y=R(x)$ on $[a,b]$ where $P, Q, R$ are continuous functions on $[a,b]$. Prove that either $y_1=y_2$ or $\{x \in [a,b] \mid y_1(x)=y_2(x)\}$ is finite.

My try : Let the set be infinite. Then, since $[a,b]$ is compact, there is a limit point $x_0$. Let, $y=y_1-y_2$. Then, at $y(x_0)=0$, $y'(x_0)=0$ and $\require{cancel} y''(x_0)=\cancel{R(x_0)}0$. I cannot go any further. Can anyone help? Hints are much more welcome than complete solutions.

Answer 1

You can go no further at this point because your last statement $y''(x_0)=R(x_0)$ is wrong. $y=y_1-y_2$ is a solution of the homogeneous equation $$y''(x)+P(x)y'(x)+Q(x)y(x)=0.$$ With initial conditions $y(x_0)=0$, $y'(x_0)=0$ this gives the zero solution.

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You could also directly conclude that

• Any root of a non-zero solution of the homogeneous equation is isolated.

• Any subset of isolated points in a compact set is finite.

Answer 2

You are almost there. A little more justification, and you can appeal to the existence and uniqueness theorem for 2nd order linear differential equations in order to finish off the problem.