Question

(D+3)^2y=sinh2x

Answer 1

Ans:-$(D+3)^2y=Sinh2x$

Auxiliary equation

$(D+3)^2=0\\$

$D=-3,-3$ are the roots

$\therefore C.F.=(C_1+C_2x)e^{-3x}$

$Sinhx=\dfrac{e^x-e^{-x}}{2}$

Particular Integral

$P.I.=\dfrac{Sinh2x}{(D+3)^2}$

$=\dfrac{1}{2}[\dfrac{e^{2x}-e^{-2x}}{(D+3)^2}]$

$=\dfrac{1}{2}\dfrac{e^{2x}}{(D+3)^2}-\dfrac{1}{2}\dfrac{e^{-2x}}{(D+3)^2}$

$=\dfrac{1}{2}\dfrac{e^{2x}}{(2+3)^2}-\dfrac{1}{2}\dfrac{e^{-2x}}{(-2+3)^2}$

$=\dfrac{e^{2x}}{50}-\dfrac{e^{-2x}}{2}$

Hence the complete solution

$Y=C.F.+P.I$

$Y=(C_1+C_2x)e^{-3x}+\ \dfrac{e^{2x}}{50}-\dfrac{e^{-2x}}{2}$