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  • Need solution for RD Sharma maths Class 12 Chapter 21 Differential Equation Exercise Multiple Choice Question Question 18 textbook solution.

Need solution for  RD Sharma maths Class 12 Chapter 21 Differential Equation Exercise Multiple Choice Question Question 18 textbook solution.

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

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

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

Explanation : \frac{d y}{d x}+P(x) y=Q(x) \text { i.e.IF }=e^{\int P(x) d x}

\begin{aligned} &I F=e^{\int \frac{1}{x} d x} \\ &=e^{\log x} \\ &=x \end{aligned}

So, the solution of differential equation is

\begin{aligned} &y x=\int x \sin x d x+C \\ &\Rightarrow y x=-x \cos x+\int \cos x d x+C \end{aligned}                        [integrating by parts]

\begin{aligned} &\Rightarrow y x=-x \cos x+\sin x+C \\ &\Rightarrow y x+x \cos x=\sin x+C \\ &\Rightarrow x(y+\cos x)=\sin x+C \end{aligned}

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