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

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

Answers (1)

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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