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Provide solution for RD Sharma maths Class 12 Chapter 21 Differential Equation Exercise Multiple Choice Question Question 6 textbook solution.

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Answer : \text { (d) } y \sin x=x+C

Hint: The differential equation is linear in y i.e. \frac{d y}{d x}+P(x) y=Q(x) \text { then } I F=e^{\int P(x) d x}

Given : \frac{d y}{d x}+y \cot x=\operatorname{cosec} x

Explanation: The given differential equation is linear in y

\begin{aligned} &\frac{d y}{d x}+P(x) y=Q(x) \text { then If }=e^{\int P(x) d x} \\ &\text { So, } I F=e^{\int \cot x d x} \\ &\qquad \begin{aligned} &=e^{\ln \sin x} \\ &=\sin x \end{aligned} \end{aligned}

So, the solution of the differential equation is

\begin{aligned} &y \sin x=\int \operatorname{cosec} x \sin x d x+C \\ &y \sin x=\int \frac{1}{\sin x} \sin x d x+C \\ &y \sin x=\int d x+C \\ &y \sin x=x+C \end{aligned}

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