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Explain solution for  RD Sharma Class 12 Chapter 21 Differential Equation Exercise Multiple Choice Question Question 25 maths textbook solution.

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Answer :  \text { (a) } \phi\left(\frac{y}{x}\right)=k x

Hint : Let y=vx

Given : \frac{d y}{d x}=\frac{y}{x}+\frac{\phi\left(\frac{y}{x}\right)}{\phi^{\prime}\left(\frac{y}{x}\right)}                         ...(i)

Explanation : \text { Let } y=v x

\Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x}

Put in (i)

\begin{aligned} &\Rightarrow v+x \frac{d v}{d x}=v+\frac{\phi(v)}{\phi^{\prime}(v)} \\ &\Rightarrow x \frac{d v}{d x}=\frac{\phi(v)}{\phi^{\prime}(v)} \end{aligned}

\begin{aligned} &\Rightarrow \frac{\phi^{\prime}(v)}{\phi(v)} d v=\frac{d x}{x} \\ &\text { Let } \phi(v)=t \\ &\Rightarrow \phi^{\prime}(v) d v=d t \end{aligned}

\Rightarrow \frac{d t}{t}=\frac{d x}{x}

Integrate both sides

\begin{aligned} &\Rightarrow \log t=\log x+\log k \\ &\Rightarrow \log t=\log k x \\ &\Rightarrow t=k x \end{aligned}

\begin{aligned} &\Rightarrow \phi(v)=k x \\ &\Rightarrow \phi\left(\frac{y}{x}\right)=k x \end{aligned}

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