solve y^(′′) +3y^′ −y =sin(2x)


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Question Number 61180 by mathsolverby Abdo last updated on 30/May/19

$s o l v e y^{^{″}} + 3 y^{^{'}} - y = s i n (2 x)$
	Answered by tanmay last updated on 30/May/19

	$\frac{d^{2} y}{{d x}^{2}} + 3 \frac{d y}{d x} - y = s i n 2 x$ $C . F d e t e r m i n a t i o n$ $\frac{d^{2} y}{{d x}^{2}} + 3 \frac{d y}{d x} - y = 0$ $y = {A e}^{m x}$ ${A m}^{2} e^{m x} + 3 {A m e}^{m x} - {A e}^{m x} = 0$ $(m^{2} + 3 m - 1) = 0 [{A e}^{m x} \neq 0]$ $m = \frac{- 3 \pm \sqrt{9 + 4}}{2} = \frac{- 3 \pm \sqrt{13}}{2}$ $C . F = C_{1} e^{(\frac{- 3 + \sqrt{13}}{2}) x} + C_{2} e^{(\frac{- 3 - \sqrt{13}}{2}) x}$ $P . I$ $(D^{2} + 3 D - 1) y = s i n 2 x$ $y = \frac{D^{2} - 1 - 3 D}{{(D^{2} - 1)}^{2} - 9 D^{2}} \times s i n 2 x [D s i n 2 x = 2 c o s 2 x a n d D^{2} s i n 2 x = - 4 s i n 2 x]$ $y = \frac{- 4 s i n 2 x - s i n 2 x - 6 c o s 2 x}{{(- 2^{2} - 1)}^{2} - 9 (- 2^{2})}$ $c o m p l e t e s o l u t i o n$ $y = C_{1} e^{(\frac{- 3 + \sqrt{13}}{2}) x} + C_{2} e^{(\frac{- 3 - \sqrt{13}}{2}) x} + \frac{- 5 s i n 2 x - 6 c o s 2 x}{25 - 36}$ $y = C_{1} e^{(\frac{- 3 + \sqrt{13}}{2}) x} + C_{2} e^{(\frac{- 3 - \sqrt{13}}{2}) + \frac{5 s i n 2 x + 6 c o s 2 x}{11}}$
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