Question

The solution of differential equation:

$\frac{d^{2}y}{d{x}^2}+4\frac{dy}{dx}+6y=3^{x}isy\left(x\right)theny\left(x\right)is$

• A. $e^{-2x}[c_{1}cos\sqrt{2}x+c_{2}sin\sqrt{2}x]+\frac{3^x}{11.6}$
• B. $e^{-2x}[c_{1}cos\sqrt{2}x+c_{2}sin\sqrt{2}x]$
• C. $e^{-\sqrt{2}x}[c_{1}cos2x+c_{2}sin2x]+\frac{3^x}{11.6}$
• D. $e^{-\sqrt{2}x}[c_{1}cos2x+c_{2}sin2x]+\frac{3^x}{9.6}$

Answer 1

The correct option is A $e^{-2x}[c_{1}cos\sqrt{2}x+c_{2}sin\sqrt{2}x]+\frac{3^x}{11.6}$

Complete Solution CS
$CS=CF+PI$

Now Auxilliary equation

$(D^2+4D+6)y=0$
$\Rightarrow m^2+4m+6=0$

$\Rightarrow m=-2\pm \sqrt{2}i$

So $C.F\to e^{-2x}[c_{1}cos\sqrt{2}x+c_{2}sin\sqrt{2}x]$ ...(1)

Now,

$PI\to \frac{3^x}{D^2+4D+6}=\frac{e^{xln3}}{D^2+4D+6}$

$\Rightarrow P.I=\frac{e^{xln3}}{(ln3{)}^2+4.ln3+6}=\frac{e^{xln3}}{11.6}=\frac{3^x}{11.6}$

$\Rightarrow C.S:y\left(x\right)=e^{-2x}[c_1\cos \sqrt{2}x+c_2\sin \sqrt{2}x]+\frac{3^x}{11.6}$