Question: Find all non-trivial solutions to the boundary value problem [ frac{d^{4} X}{d x^{4}}-lambda X=0 quad X(0)=0 quad X^{prime prime}(0)=0 quad X(1)=0 quad X^{prime prime}(1)=0 ]

Find all non-trivial solutions to the boundary value problem

\[ \frac{d^{4} X}{d x^{4}}-\lambda X=0 \quad X(0)=0 \quad X^{\prime \prime}(0)=0 \quad X(1)=0 \quad X^{\prime \prime}(1)=0 \]

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