Please solve RD Sharma Class 12 Chapter 21 Differential Equation Exercise Revision Exercise (RE) Question 52 maths textbook solution.

Answer : $y=4(x-1)+c e^{-x}$

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

Given     :  $\frac{dy}{dx}+y=4x$

Solution : $\frac{dy}{dx}+y=4x$                                           .....(i)

Compare with, $\frac{dy}{dx}+py=Q$

Where, $P=1, Q=4x$

Therefore,

\begin{aligned} &\text { I.F }=e^{\int P d x} \\ &\qquad=e^{\int d x} \\ &\; \; \; \; \; \; \; =e^{x} \end{aligned}

Hence, the solution is ,

\begin{aligned} y \times I . F &=\int(I . F \times Q) d x+c \\ y \quad e^{x} &=\int e^{x} 4 x d x+c \end{aligned}

Integrating by parts,

\begin{aligned} &y e^{x}=4 x \int e^{x} d x-4 \int\left[\frac{d}{d x}(x) \int e^{x} d x\right]+c \\ &y e^{x}=4 x e^{x}-4 \int e^{x} d x+c \\ &y e^{x}=4 x e^{x}-4 e^{x}+c \end{aligned}

\begin{aligned} y e^{x} &=4(x-1) e^{x}+c \\ y &=4(x-1)+c e^{-x} \end{aligned}                       is required solution