Problem 10
Fill up the blanke or chose the correct answer in the following problems: $$ \frac{d}{d x}\left[x^{N} J_{n}(x)\right]=\ldots \ldots $$
Problem 21
Legendre's polynomial of first degree \(=x\).
Problem 23
\(x=0\) is a regular singular point of \(2 x^{2} y^{\prime}+3 x y^{\prime}+\left(x^{2}-4\right) y=0\).
Problem 25
If \(J_{6}\) and \(J_{1}\) are Bessel functions, then \(J_{1}^{\prime}(x)\) is given by \((a)-\bar{J}_{0}\) (b) \(J_{6}(x)-1 / x J_{1}(x)\) (c) \(J_{0}(x)+\frac{1}{x} J_{1}(x)\).
Problem 28
The series \(x-\frac{x^{3}}{2^{2}(1)^{2}}+\frac{x^{5}}{2^{4}(2 !)^{2}}-\frac{x^{7}}{2^{6}(3 \mid)^{2}}+\ldots . .\) m equals \(\left(\right.\) a) \(\delta_{1 / 2}(x)\) (b) \(J_{0}(x)\) (c) \(x_{v} J_{0}(x)\) \((d) x J_{12}(x) .\)
Problem 29
If \(\int_{-1}^{1} P_{n}(x) d x=2\), then \(n\) is (a) 0 (b) 1 (c) \(-1\) (d) none of these.
Problem 30
The value of \(\int_{-1}^{1}(2 x+1) P_{3}(x) d x\) where \(P_{3}(x)\) is the third degree Legendre polynomial, is (a) \(\mathbf{1}\) (b) \(-1\) (c) 2 (d) \(0 .\)
Problem 31
The value of the integral \(\int_{-1}^{1} x^{3} P_{3}(x) d x\), where \(P_{3}(x)\) is a Legendre polynomial of degrre 3 , is (a) 0 (b) \(\frac{2}{35}\) (c) \(\frac{4}{35}\) (d) \(\frac{11}{35}\).
Problem 32
The polynomial \(2 x^{2}+x+3\) in terms of Legendre polynomials in (a) \(\frac{1}{3}\left(4 P_{2}-3 P_{1}+11 P_{0}\right)\) (b) \(\frac{1}{8}\left(4 P_{2}+3 P_{1}-11 P_{0}\right)\) (c) \(\frac{1}{3}\left(4 P_{2}+3 P_{1}+11 P_{0}\right)\) (d) \(\frac{1}{3}\left(4 P_{2}-3 P_{1}-11 P_{0}\right)\)
Problem 34
Legendre polynomial \(P_{\mathrm{b}}(x)=\lambda\left(63 x^{5}-70 x^{3}+15 x\right)\) where \(\lambda\) is equal to (a) \(1 / 2\) (b) \(1 / 5\) (c) \(1 / 8\) (d) \(1 / 10\).
