Question

The solution of the differential equation $\frac{d^{2}y}{d{x}^2}+6\frac{dy}{dx}+9y=9x+6$ with $C_1$ and $C_2$ as constant is

• A. $y=(C_{1}x+C_2)e^{-3x}$
• B. $b=C_1{e}^{3x}+C_2{e}^{-3x}$
• C. $y=(C_{1}x+C_2)e^{-3x}+x$
• D. $y=(C_{1}x+C_2)e^{3x}+x$

Answer 1

The correct option is C $y=(C_{1}x+C_2)e^{-3x}+x$

Given DE is
$(D^2+6D+9)y=9x+6$ ... $\left(1\right)$
A.E. is $m^2+6m+9=0$
$m=-3,-3$
So, C.F. $=(C_{1}x+C_2)e^{-3x}$
Now, P.I. = $\frac{1}{f\left(D\right)}\left(Q\right(x\left)\right)$
$=\frac{1}{(D+3{)}^2}(9x+6)$
$=\frac{1}{9}[1-2\frac{D}{3}+3\left(\frac{D^2}{9}\right)-.....](9x+6)$
$=\frac{1}{9}(9x+6)-\frac{2}{3}=x+\frac{2}{3}-\frac{2}{3}=x$
General solution is

$y=C.F.+P.I.$
$\therefore$ $y=(C_{1}x+C_2)e^{-3x}+x$