x′′ − (5/t) x′ + (8/t^2 ) x = ((2 ln t)/t^2 ) solve the differential eqn


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Question Number 184683 by alcohol last updated on 10/Jan/23

$x^{″} - \frac{5}{t} x^{'} + \frac{8}{t^{2}} x = \frac{2 l n t}{t^{2}}$ $s o l v e t h e d i f f e r e n t i a l e q n$
	Answered by qaz last updated on 10/Jan/23

	$t^{2} x^{″} - 5 {t x}^{'} + 8 x = 2 l n t$ $t = e^{z},$ $x_{p} = \frac{1}{D (D - 1) - 5 D + 8} 2 z = (\frac{1}{D - 4} - \frac{1}{D - 2}) z$ $= e^{4 z} \frac{1}{D} e^{- 4 z} z - e^{2 z} \frac{1}{D} e^{- 2 z} z = e^{4 z} \int e^{- 4 z} z d z - e^{2 z} \int e^{- 2 z} z d z$ $= \frac{1}{4} z + \frac{3}{16}$ $\Rightarrow x = C_{1} e^{4 z} + C_{2} e^{2 z} + \frac{1}{4} z + \frac{3}{16} = C_{1} t^{4} + C_{2} t^{2} + \frac{1}{4} l n t + \frac{3}{16}$
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