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I have difficulties to resolve the following problem. Thank you in advance to help me please

Let $y_1$ and $y_2$ two independents solutions of the problem $$ \begin{cases} a(x) y''+ b(x) y' + c(x) y=0\\[0.5em] l_1(y)= a_0 y(\alpha)+ a_1y'(\alpha)+ b_0 y(\beta)+ b_1 y'(\beta)=0\\ l_2(y)= c_0 y(\alpha)+ c_1 y'(\alpha)+ d_0 y(\beta)+ d_1 y'(\beta)=0 \end{cases} $$ Prove that this problem admits an non trivial solution if and only if $$l_1(y_1)·l_2(y_2)-l_1(y_2)·l_2(y_1)\neq 0.$$

What we deduce.

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Sketch of proof:

Note that $l_1(y)$ is linear in $y$. That means $l_1(A\;y_1 + B\;y_2) = A\;l_1(y_1) + B\;l_1(y_2)$. The same is true of $l_2(y)$. In general, a solution to the differential equation will take the form $y = A\;y_1 + B\;y_2$, and $l_1(y) = 0$ and $l_2(y) = 0$ then form a system of linear equations in $A$ and $B$. Find the condition that that equation must satisfy to have nontrivial solutions.

I think you should be able to fill in the blanks.

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