Find-y-CF-PI-in-following-differential-equation-d-2-y-dx-2-3-dy-dx-2y-e-2x-sinx-

Question Number 86242 by niroj last updated on 27/Mar/20

Find y = C F + P I in following differential equation :

\frac{d^{2} y}{{dx}^{2}} + 3 \frac{dy}{dx} + 2 y = e^{2 x} sinx .

Answered by TANMAY PANACEA. last updated on 27/Mar/20

y = e^{m x}

m^{2} e^{m x} + 3 {m e}^{m x} + 2 e^{m x} = 0

e^{m x} (m + 1) (m + 2) = 0

e^{m x} \neq 0 s o m = - 1, - 2

C . F

{A e}^{- x} + {B e}^{- 2 x}

P . I = \frac{e^{2 x} s i n x}{D^{2} + 3 D + 2}

= e^{2 x} . \frac{s i n x}{{(D + 2)}^{2} + 3 (D + 2) + 2}

= e^{2 x} . \frac{s i n x}{D^{2} + 7 D + 12}

= e^{2 x} . \frac{D^{2} + 12 - 7 D}{{(D^{2} + 12)}^{2} - 49 D^{2}} . s i n x

= e^{2 x} . \frac{- s i n x + 12 s i n x - 7 c o s x}{{(- 1^{2} + 12)}^{2} - 49 (- 1^{2})}

= e^{2 x} . \frac{11 s i n x - 7 c o s x}{121 + 49} = e^{2 x} . \frac{11 s i n x - 7 c o s x}{170}

y = {A e}^{- x} + {B e}^{- 2 x} + e^{2 x} . \frac{11 s i n x - 7 c o s x}{170}

Commented by niroj last updated on 27/Mar/20

great job sir .

Commented by TANMAY PANACEA. last updated on 27/Mar/20

t h a n k y o u s i r