Question

The differential equation whose general solution is given by, y =
$\left(c_{1}cos\right(x+c_2\left)\right)-\left(c_3{e}^{-x+c_4}\right)+\left(c_{5}sinx\right)$, where $c_1,c_2,c_3,c_4,c_5$ are arbitrary constants, is

• A. $\frac{d^{4}y}{d{x}^4}-\frac{d^{2}y}{d{x}^2}+y=0$
• B. $\frac{d^{3}y}{d{x}^3}+\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}+y=0$
• C. $\frac{d^{5}y}{d{x}^5}+y=0$
• D. $\frac{d^{3}y}{d{x}^3}-\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}-y=0$

Answer 1

The correct option is B $\frac{d^{3}y}{d{x}^3}+\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}+y=0$

$y=c_{1}cos(x+c_2)-\left(c_3{e}^{-x+c_4}\right)+\left(c_{5}sinx\right)\Rightarrow y=c_1(cosxcos{c}_2-sinxsin{c}_2)-\left(c_3{e}^{c_4}{e}^{-x}\right)+\left(c_{5}sinx\right)\Rightarrow y=\left(c_{1}cos{c}_2\right)cosx-(c_{1}sin{c}_2-c_5)sinx-\left(c_3{e}^{c_4}\right)e^{-x}\Rightarrow y=lcosx+msinx-n{e}^{-x}............\left(1\right)$
where l, m, n are arbitrary constant
$\Rightarrow \frac{dy}{dx}=-lsinx+mcosx+n{e}^{-x}..............\left(2\right)\Rightarrow \frac{d^{2}y}{d{x}^2}=-lcosx-msinx-n{e}^{-x}.............\left(3\right)\Rightarrow \frac{d^{3}y}{d{x}^3}=lsinx-mcosx+n{e}^{-x}..............\left(4\right)Fromequations\left(1\right)+\left(3\right),\frac{d^{2}y}{d{x}^2}+y=-2n{e}^{-x}.............\left(5\right)Fromequations\left(2\right)+\left(4\right),\frac{d^{3}y}{d{x}^3}+\frac{dy}{dx}=2n{e}^{-x}.............\left(6\right)Fromequations\left(5\right)+\left(6\right),weget\frac{d^{3}y}{d{x}^3}+\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}+y=0$