# Get Answers to all your Questions

#### Please solve RD Sharma Class 12 Chapter 21 Differential Equation Exercise Revision Exercise (RE) Question 51 maths textbook solution.

Answer :    $y=\frac{1}{13}(2 \sin 3 x-3 \cos 3 x)+C e^{-2 x}$

Hint               : You must know the rules of solving differential equation and integration.

Given            :      $\frac{d y}{d x}+2 y=\sin 3 x$

Solution        :    $\frac{d y}{d x}+2 y=\sin 3 x$

Compare with,

$\frac{d y}{d x}+P y=Q$

where, $P=2$ and $Q= \sin 3x$

Therefore,

\begin{aligned} \text { Integrating factor }=\text { I.F } &=e^{\int p d x} \\ &=e^{\int 2 d x} \\ &=e^{2 x} \end{aligned}

The solution is,

\begin{aligned} &y \times I . F=\int(Q \times I . F) d x+C \\ &y e^{2 x}=\int e^{2 x} \times \sin 3 x d x+C \\ &y e^{2 x}=I+C \end{aligned}           ......(i)

Where,    $I=\int e^{2 x} \sin 3 x d x$                    ......(ii)

Apply integrating by parts,

\begin{aligned} I &=e^{2 x} \int \sin 3 x d x-\int\left[\frac{d e^{2 x}}{d x} \int \sin 3 x d x\right] d x \\ I &=\frac{-e^{2 x} \cos 3 x}{3}+\frac{2}{3} \int e^{2 x} \cos 3 x d x \\ I &=\frac{-e^{2 x} \cos 3 x}{3}+\frac{2}{3}\left[e^{2 x} \int \cos 3 x d x-\int\left\{\frac{d e^{2 x}}{d x} \int \cos 3 x d x\right\} d x\right] \end{aligned}

\begin{aligned} I &=\frac{-e^{2 x} \cos 3 x}{3}+\frac{2}{3}\left[\frac{e^{2 x} \sin 3 x}{3}-\frac{2}{3} \int e^{2 x} \sin 3 x d x\right] \\ I &=\left[\frac{2}{9} e^{2 x} \sin 3 x-\frac{e^{2 x} \cos 3 x}{3}\right]-\frac{4}{9} I \end{aligned}

\begin{aligned} I+\frac{4}{9} I &=e^{2 x}\left(\frac{2}{9} \sin 3 x-\frac{\cos 3 x}{3}\right) \\ \frac{13}{9} I &=e^{2 x}\left(\frac{2}{9} \sin 3 x-\frac{\cos 3 x}{3}\right) \\ I &=\frac{9}{13} e^{2 x}\left(\frac{2}{9} \sin 3 x-\frac{\cos 3 x}{3}\right) \\ I &=\frac{1}{13} e^{2 x}(2 \sin 3 x-3 \cos 3 x)+C \end{aligned}                                         .....(iii)

From (i) and (iii), we get,

\begin{aligned} y e^{2 x} &=\frac{e^{2 x}}{13}(2 \sin 3 x-3 \cos 3 x)+C \\ y \quad &=\frac{1}{13}(2 \sin 3 x-3 \cos 3 x)+C e^{-2 x} \end{aligned}     Is required solution.