Question

The particular integral of the differential equation $\frac{d^{2}y}{d{x}^2}-\frac{2dy}{dx}=e^{2x}sinx$

• A. $\frac{-e^{2x}}{10}(3\cos x-2\sin x)$
• B. $\frac{e^{2x}}{10}(3\cos x-2\sin x)$
• C. $\frac{-e^{2x}}{5}(2\cos x+\sin x)$
• D. $\frac{e^{2x}}{5}(2\cos x-\sin x)$

Answer 1

The correct option is C $\frac{-e^{2x}}{5}(2\cos x+\sin x)$

Finding the particular integral (P.I):

$\frac{d^{2}y}{d{x}^2}-\frac{2dy}{dx}=e^{2}xsinx$

$P.I.=\frac{e^{2x}\sin x}{D^2-2D}$

$=e^{2x}\cdot \frac{1}{(D+2{)}^2-2(D+2)}\cdot \sin x$

$=e^{2x}\cdot \frac{1}{D^2+2D}\cdot \sin x$

$=e^{2x}\cdot \frac{1}{(-1{)}^2+2D}\cdot sinx$

$=e^{2x}\cdot \frac{1}{2D-1}\cdot sinx$

$=e^{2x}\cdot \frac{(2D+1)}{(4D^2-1)}\cdot sinx$

$=e^{2x}\cdot \frac{(2\frac{d}{dx}sin+sinx)}{4(-1^2)-1}$

$P.I.=\frac{e^{2x}\cdot (2\cos x+\sin x)}{-5}$


$P.I.=-\frac{e^{2x}}{5}$(2 cos x + sin x)