#### Provide Solution for RD Sharma Class 12 Chapter 21 Differential Equation Exercise 21.10 Question 36 subquestion (vii)

Answer:  \begin{aligned} &y=\frac{1}{2} e^{\sin x}+\frac{c}{e^{\sin x}} \\ & \end{aligned}

Give:  $\frac{d y}{d x}+y \cos x=e^{\sin x} \cos x \\$

Hint: Using   $\int \cos x d x \\$

Explanation:  $\frac{d y}{d x}+\cos x y=e^{\sin x} \cos x$

This is a first order linear differential equation of the form

\begin{aligned} &\frac{d y}{d x}+P y=Q \\ &P=\cos x \text { and } Q=e^{\sin x} \cos x \end{aligned}

The integrating factor  $If$ of this differential equation is

\begin{aligned} &I f=e^{\int \cos x d x} \\ &=e^{\sin x} \qquad\left[\int \sin x d x=\cos x+C\right] \end{aligned}

Hence, the solution of different equation is

\begin{aligned} &y I f=\int Q I f d x+C \\ &=y e^{\sin x}=\int e^{\sin x} \cos x e^{\sin x} d x+C \end{aligned}

\begin{aligned} &=y e^{\sin x}=\int e^{2 \sin x} \cos x d x+C \\ &=y e^{\sin x}=\int e^{2 t} d t+C \|[\sin x=t, \cos x d x=d t] \\ &=y e^{\sin x}=\left[\frac{e^{2 t}}{2}\right]+C \end{aligned}

\begin{aligned} &=y e^{\sin x}=\frac{e^{2 \sin x}}{2}+C \\ &=y=\frac{e^{2 \sin x}}{2 e^{\sin x}}+\frac{C}{e^{\sin x}} \\ &=y=\frac{e^{\sin x}}{2}+\frac{C}{e^{\sin x}} \end{aligned}