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Please solve RD Sharma Class 12 Chapter 21 Differential Equation Exercise Multiple Choice Question Question 1 maths textbook solution.

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Answer : (c) \log\; x

Hint :  \frac{d y}{d x}+P(x) y=Q(x) \text { then } I F=e^{\int P(x) d x}

Given : (x \log x) \frac{d y}{d x}+y=2 \log x

                 \begin{aligned} &\frac{d y}{d x}+\frac{y}{x \log x}=\frac{2 \log x}{x \log x} \\ \end{aligned}

                \begin{aligned} &\frac{d y}{d x}+\frac{y}{x \log x}=\frac{2}{x} \end{aligned}

The given differential equation is linear in y

\begin{aligned} &I f=e^{\int P(x) d x} \\ &I F=e^{\int \frac{1}{x \log x} d x} \\ &\text { Let } \log x=t \end{aligned}

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

\begin{aligned} &=t \\ &I F=\log x \end{aligned}

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