Given that \( U_{n}=\{x(1-x)\}^{n} \) and \( n \geq 2 \) and
\[
\frac{d^{2} U_{n}}{d x^{2}}=n(n-... VIDEO
Given that \( U_{n}=\{x(1-x)\}^{n} \) and \( n \geq 2 \) and
\[
\frac{d^{2} U_{n}}{d x^{2}}=n(n-1) U_{n-2}-2 n(2 n-1) U_{n-1}
\]
further if \( V_{n}=\int_{0}^{1} e^{x} \cdot U_{n} d x \), for \( n \geq 2 \), Prove that
\[
V_{n}+2 n(2 n-1) V_{n-1}-n(n-1) V_{n-1}=0
\]
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