Solve the d.e using method of variation of parameter. (d^2 y/dx^2 )+3(dy/dx)+2y=sin(e^x )


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Question Number 47624 by Umar last updated on 12/Nov/18

$Solve the d . e using method of variation$ $of parameter .$ $\frac{d^{2} y}{{dx}^{2}} + 3 \frac{dy}{dx} + 2 y = \sin (e^{x})$
	Answered by tanmay.chaudhury50@gmail.com last updated on 12/Nov/18
	$let t=e^x t=e^x (dt/dx)=e^x =t (dy/dx)=(dy/dt)×(dt/dx)=t(dy/dt) (d/dx)((dy/dx))=(d/dt)(t(dy/dt))×(dt/dx) =t×{t(d^2 y/dt^2 )+(dy/dt)}=t^2 (d^2 y/dt^2 )+t(dy/dt) now (d^2 y/dx^2 )+3(dy/dx)+2y=sin(e^x ) t^2 (d^2 y/dt^2 )+t(dy/dt)+3t(dy/dt)+2y=sin(t) t^2 (d^2 y/dt^2 )+4t(dy/dt)+2y=sint now [θ(θ−1)+4θ+2]y=sint θ=(d/dt) for C.F θ^2 −θ+4θ+2=0 (θ+1)(θ+2)=0 θ=−1,−2 so C.F=C_1 e^(−t) +C_2 e^(−2t) →C_1 e^(−e^x ) +C_2 e^(−2e^x ) P.I y=(1/(θ^2 +3θ+2))×sint =((θ^2 +2−3θ)/((θ^2 +2+3θ)(θ^2 +2−3θ)))×sint =((θ^2 +2−3θ)/((θ^2 +2)^2 −9θ^2 ))sint =(1/((−1^2 +2)^2 −9(−1^2 )))×{−sint+2sint−cost} =(1/(1+9))×{sint−cost} =(1/(10))×{sin(e^x )−cos(e^x )} so complete solution is C_1 e^(−e^x ) +C_2 e^(−2e^x ) +(1/(10)){sin(e^x )−cos(e^x )} pls check...$
	$l e t$ $t = e^{x}$ $t = e^{x}$ $\frac{d t}{d x} = e^{x} = t$ $\frac{d y}{d x} = \frac{d y}{d t} \times \frac{d t}{d x} = t \frac{d y}{d t}$ $\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d t} (t \frac{d y}{d t}) \times \frac{d t}{d x}$ $= t \times {t \frac{d^{2} y}{{d t}^{2}} + \frac{d y}{d t}} = t^{2} \frac{d^{2} y}{{d t}^{2}} + t \frac{d y}{d t}$ $n o w$ $\frac{d^{2} y}{{d x}^{2}} + 3 \frac{d y}{d x} + 2 y = s i n (e^{x})$ $t^{2} \frac{d^{2} y}{{d t}^{2}} + t \frac{d y}{d t} + 3 t \frac{d y}{d t} + 2 y = s i n (t)$ $t^{2} \frac{d^{2} y}{{d t}^{2}} + 4 t \frac{d y}{d t} + 2 y = s i n t$ $n o w$ $[θ (θ - 1) + 4 θ + 2] y = s i n t θ = \frac{d}{d t}$ $f o r$ $C . F$ $θ^{2} - θ + 4 θ + 2 = 0$ $(θ + 1) (θ + 2) = 0$ $θ = - 1, - 2$ $s o C . F = C_{1} e^{- t} + C_{2} e^{- 2 t} \to C_{1} e^{- e^{x}} + C_{2} e^{- 2 e^{x}}$ $P . I$ $y = \frac{1}{θ^{2} + 3 θ + 2} \times s i n t$ $= \frac{θ^{2} + 2 - 3 θ}{(θ^{2} + 2 + 3 θ) (θ^{2} + 2 - 3 θ)} \times s i n t$ $= \frac{θ^{2} + 2 - 3 θ}{{(θ^{2} + 2)}^{2} - 9 θ^{2}} s i n t$ $= \frac{1}{{(- 1^{2} + 2)}^{2} - 9 (- 1^{2})} \times {- s i n t + 2 s i n t - c o s t}$ $= \frac{1}{1 + 9} \times {s i n t - c o s t}$ $= \frac{1}{10} \times {s i n (e^{x}) - c o s (e^{x})}$ $s o c o m p l e t e s o l u t i o n i s$ $C_{1} e^{- e^{x}} + C_{2} e^{- 2 e^{x}} + \frac{1}{10} {s i n (e^{x}) - c o s (e^{x})}$ $p l s c h e c k . . .$
	Commented by Umar last updated on 12/Nov/18

	$thanks$
	Commented by tanmay.chaudhury50@gmail.com last updated on 12/Nov/18

	$m o s t w e l c o m e . . .$
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