find the general solution y(t) of the ordinary differential equation y′′ + ω^2 y=cos ωt ,where w>0


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Question Number 59800 by necx1 last updated on 14/May/19

$f i n d t h e g e n e r a l s o l u t i o n y (t) o f t h e$ $o r d i n a r y d i f f e r e n t i a l e q u a t i o n$ $y^{″} + ω^{2} y = \cos ω t, w h e r e w > 0$
	Answered by MJS last updated on 15/May/19

	$1^{st} attempt$ $y = C_{1} \cos ω t + C_{2} \sin ω t$ $y^{″} = - C_{1} ω^{2} \cos ω t - C_{2} ω^{2} \sin ω t$ $y^{″} + ω^{2} y = 0$ $2^{nd} attempt$ $y = C_{1} \cos ω t + C_{2} \sin ω t + C_{3} t \sin w t$ $y^{″} = - C_{1} ω^{2} \cos ω t - C_{2} ω^{2} \sin ω t + 2 C_{3} ω \cos ω t - C_{3} ω^{2} t \sin ω t$ $y^{″} + ω^{2} y = 2 C_{3} ω \cos ω t$ $\Rightarrow C_{3} = \frac{1}{2 ω}$ $\Rightarrow y (t) = C_{1} \cos ω t + (\frac{t}{2 ω} + C_{2}) \sin ω t$
	Answered by tanmay last updated on 15/May/19

	$\frac{d^{2} y}{{d t}^{2}} + w^{2} y = c o s w t$ $(D^{2} + w^{2}) y = c o s w t$ $C . F d e t e r m i n a t i o n$ $\frac{d^{2} y}{{d t}^{2}} + w^{2} y = 0$ $l e t y = {A e}^{α t} i s a s o l u t i o n$ $\frac{d y}{d t} = A α e^{α t}$ $\frac{d^{2} y}{{d t}^{2}} = A α^{2} e^{α t}$ $A α^{2} e^{α t} + w^{2} {A e}^{α t} = 0$ ${A e}^{α t} (α^{2} + w^{2}) = 0$ ${A e}^{α t} \neq 0$ $α^{2} + w^{2} = 0$ $α = \pm i w$ $C . F$ $y = C_{1} e^{i w t} + C_{2} e^{- i w t}$ $P a r t i c u l a r i n t r e g a l$ $y = \frac{c o s w t}{D^{2} + w^{2}}$ $p = \frac{c o s w t}{D^{2} + w^{2}}$ $q = \frac{s i n w t}{D^{2} + w^{2}}$ $p + i q = \frac{e^{i w t}}{(D + i w) (D - i w)}$ $p + i q = \frac{e^{i w t}}{(i w + i w)} \times \frac{1}{D + i w - i w}$ $p + i q = \frac{e^{i w t}}{2 i w} \times x [\frac{1}{D} \times 1 = x]$ $p + i q = \frac{x (c o s w t + i s i n w t)}{2 i w}$ $= \frac{x c o s w t}{2 w} \times \frac{i}{- 1} + \frac{x s i n w t}{2 w}$ $p + i q = (\frac{x s i n w t}{2 w}) + i (\frac{- x c o s w t}{2 w})$ $p = \frac{- x c o s w t}{2 w} \to (t a k i n g t h e r e a l p a r t)$ $c o m p l t e s o l u t i o n$ $y = C_{1} e^{i w t} + C_{2} e^{- i w t} + (\frac{- x c o s w t}{2 w})$ $p l s c h e c k t h e a n s w e r$
	Commented bynecx1 last updated on 15/May/19

	$h m m m . . . . I t h i n k I n e e d t o s t u d y t h i s$ $m o r e$
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