Question

Find the general solution in powers of z of the differential equation $$(z^2 - 1)y'' + 4zy' + 2y = 0$$ Assume the form $$y(z) = \sum_{n=0}^{\infty} c_n z^n$$ Then $$y'(z) = \sum_{n=1}^{\infty} n c_n z^{n-1}$$ $$y''(z) = \sum_{n=2}^{\infty} n(n-1) c_n z^{n-2}$$ $$z^2 y''(z) = \sum_{n=2}^{\infty} n(n-1) c_n z^n$$ $$-y''(z) = \sum_{n=0}^{\infty} -(n+2)(n+1) c_{n+2} z^n$$ (Note: shift of index of summation must be used here) $$4zy'(z) = \sum_{n=1}^{\infty} 4n c_n z^n$$ $$2y(z) = \sum_{n=0}^{\infty} 2 c_n z^n$$ Then $$(z^2 - 1)y'' + 4zy' + 2y$$ $$= \sum_{n=0}^{\infty} [-(n+2)(n+1) c_{n+2} + (n(n-1) + 4n + 2) c_n] z^n$$ Requiring that the terms of this series for $$(z^2 - 1)y'' + 4zy' + 2y$$ vanish gives the recurrence relation $$c_{n+2} = \frac{n(n-1) + 4n + 2}{(n+2)(n+1)} c_n$$ for $$n = 0, 1, 2, ...$$

Name: find the general solution in powers of z of the differential equation z2 1y 4zy 2y 0 assume the form yz sumn0infty cn zn then yz sumn1infty n cn zn 1 yz sumn2infty nn 1 cn zn 2 z2 yz s 32946
Uploaded: 2022-06-16T10:15:12-08:00
Duration: 10 min 25 s
Channel: Shaiju T
Description: find the general solution in powers of z of the differential equation z2 1y 4zy 2y 0 assume the form yz sumn0infty cn zn then yz sumn1infty n cn zn 1 yz sumn2infty nn 1 cn zn 2 z2 yz s 32946

          Find the general solution in powers of z of the differential equation
$$(z^2 - 1)y'' + 4zy' + 2y = 0$$
Assume the form
$$y(z) = \sum_{n=0}^{\infty} c_n z^n$$
Then
$$y'(z) = \sum_{n=1}^{\infty} n c_n z^{n-1}$$
$$y''(z) = \sum_{n=2}^{\infty} n(n-1) c_n z^{n-2}$$
$$z^2 y''(z) = \sum_{n=2}^{\infty} n(n-1) c_n z^n$$
$$-y''(z) = \sum_{n=0}^{\infty} -(n+2)(n+1) c_{n+2} z^n$$
(Note: shift of index of summation must be used here)
$$4zy'(z) = \sum_{n=1}^{\infty} 4n c_n z^n$$
$$2y(z) = \sum_{n=0}^{\infty} 2 c_n z^n$$
Then $$(z^2 - 1)y'' + 4zy' + 2y$$
$$= \sum_{n=0}^{\infty} [-(n+2)(n+1) c_{n+2} + (n(n-1) + 4n + 2) c_n] z^n$$
Requiring that the terms of this series for $$(z^2 - 1)y'' + 4zy' + 2y$$ vanish gives the recurrence relation
$$c_{n+2} = \frac{n(n-1) + 4n + 2}{(n+2)(n+1)} c_n$$ for $$n = 0, 1, 2, ...$$

find the general solution in powers of z of the differential equation z2 1y 4zy 2y 0 assume the form yz sumn0infty cn zn then yz sumn1infty n cn zn 1 yz sumn2infty nn 1 cn zn 2 z2 yz s 32946

Added by Sarah P.

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