Question

Answers the following questions: 1) Show that $\int_0^1 x^2 P_{n+1}P_{n-1}dx = \frac{n(n+1)}{(2n+3)(4n^2-1)}$ 2) Evaluate: $\int_{-1}^1 x^3 P_4(x)dx$ (Hint: $P_3(x) = \frac{1}{2}(5x^3 - 3x)$) 3) Solve in terms of Bessel functions: $x^2y'' + \frac{d}{dx}(xy) + (x^2 - \frac{5}{4})y = 0$, $y(\frac{\pi}{2}) = 0$, $y'(\frac{\pi}{2}) = -1$ 4) Show that $\int_0^{\pi} \sqrt{\pi x} J_{\frac{1}{2}}(x)dx = \sqrt{2}$ 5) Show that $y = x^n J_n(x)$ is a solution of the differential equation: $xy'' + (1 - 2n)y' + xy = 0

          Answers the following questions:
1) Show that $\int_0^1 x^2 P_{n+1}P_{n-1}dx = \frac{n(n+1)}{(2n+3)(4n^2-1)}$
2) Evaluate: $\int_{-1}^1 x^3 P_4(x)dx$ (Hint: $P_3(x) = \frac{1}{2}(5x^3 - 3x)$)
3) Solve in terms of Bessel functions:
$x^2y'' + \frac{d}{dx}(xy) + (x^2 - \frac{5}{4})y = 0$, $y(\frac{\pi}{2}) = 0$, $y'(\frac{\pi}{2}) = -1$
4) Show that $\int_0^{\pi} \sqrt{\pi x} J_{\frac{1}{2}}(x)dx = \sqrt{2}$
5) Show that $y = x^n J_n(x)$ is a solution of the differential equation:
$xy'' + (1 - 2n)y' + xy = 0

Answers the following questions:
1) Show that ∫0^1 x^2 Pn+1Pn-1dx = (n(n+1))/((2n+3)(4n^2-1))
2) Evaluate: ∫-1^1 x^3 P4(x)dx (Hint: P3(x) = (1)/(2)(5x^3 - 3x))
3) Solve in terms of Bessel functions:
x^2y” + (d)/(dx)(xy) + (x^2 - (5)/(4))y = 0, y((π)/(2)) = 0, y'((π)/(2)) = -1
4) Show that ∫0^π√(π x) J(1)/(2)(x)dx = √(2)
5) Show that y = x^n Jn(x) is a solution of the differential equation:
xy” + (1 - 2n)y' + xy = 0

Added by Jose Manuel P.

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