The eigenvalues $\lambda_{n}$ and eigenfunctions $u_{n}(x)$ of the Sturm-Liouville problem $\frac{d^{2} y}{d x^{2}}+k^{2} \lambda y=0, \quad 0 $are given by:

$(\mathrm{A}) \lambda_{n}=n^{2} \pi^{2} ; \quad u_{n}(x)=\sin \lambda_{n} x, \quad n=0,\pm 1,\pm 2, \cdots, \infty$

(B) $\lambda_{n}=n^{2} \pi^{2} / k^{2} ; \quad u_{n}(x)=\sin k n \pi x, \quad n=0,\pm 1,\pm 2, \cdots, \infty$

(C) $\lambda_{n}=n^{2} \pi^{2} / k^{2} ; \quad u_{n}(x)=\sin n \pi x, \quad n=0,\pm 1,\pm 2, \cdots, \infty$

(D) $\lambda_{n}=n^{2} \pi^{2} ; \quad u_{n}(x)=\sin n \pi x, \quad n=0,\pm 1,\pm 2, \cdots, \infty$

The eigenvalues $\lambda_{n}$ and eigenfunctions $u_{n}(x)$ of the Sturm-Liouville problem $\frac{d^{2} y}{d x^{2}}+k^{2} \lambda y=0, \quad 0 $are given by:

$(\mathrm{A}) \lambda_{n}=n^{2} \pi^{2} ; \quad u_{n}(x)=\sin \lambda_{n} x, \quad n=0,\pm 1,\pm 2, \cdots, \infty$

(B) $\lambda_{n}=n^{2} \pi^{2} / k^{2} ; \quad u_{n}(x)=\sin k n \pi x, \quad n=0,\pm 1,\pm 2, \cdots, \infty$

(C) $\lambda_{n}=n^{2} \pi^{2} / k^{2} ; \quad u_{n}(x)=\sin n \pi x, \quad n=0,\pm 1,\pm 2, \cdots, \infty$

(D) $\lambda_{n}=n^{2} \pi^{2} ; \quad u_{n}(x)=\sin n \pi x, \quad n=0,\pm 1,\pm 2, \cdots, \infty$

1 Answer

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(C) $\lambda_{n}=n^{2} \pi^{2} / k^{2} ; \quad u_{n}(x)=\sin n \pi x, \quad n=0,\pm 1,\pm 2, \cdots, \infty$

3-point Gaussian integration formula is given by:

$\int_{-1}^{1} f(x) d x \approx \sum_{j=1}^{3} A_{j} f\left(x_{j}\right)$ with $x_{1}=0, x_{2}=-x_{3}=-\sqrt{\frac{3}{5}} ; A_{1}=\frac{8}{9}, A_{2}=A_{3}=\frac{5}{9}$

This formula exactly integrates

(A) $f(x)=5-x^{7}$

(B) $f(x)=2+3 x+6 x^{4}$

(C) $f(x)=13+6 x^{3}+x^{6}$

(D) $f(x)=e^{-x^{2}}$

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