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Please Solve R.D.Sharma class 12 Chapter 21  Differential Equations Exercise 21.9 Question 10 Maths textbook Solution.

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Answer: e^{x/y}=log\: y+c.

Given:ye^{x/y}dx=\left ( xe^{x/y}+y \right )dy\:ye^{x/y}dx=\left ( xe^{x/y}+y \right )dy\:

To solve: we have to solve the given differential equation

Hint: In homogeneous differential equation put   y=vx  and  \frac{dy}{dx}=v+x\frac{dv}{dx}

Solution: we have,

ye^{x/y}dx=\left ( xe^{x/y}+y \right )dy

\Rightarrow \frac{dy}{dx}=\frac{xe^{x/y}+y}{ye^{x/y}}

It is a homogeneous equation.

Put  x=vy and \frac{dy}{dx}=v+y\frac{dv}{dy}


\begin{aligned} &v+y \frac{d v}{d y}=\frac{v y e^{v y} / y+y}{y e^{v y} / y} \\ &\Rightarrow v+y \frac{d v}{d y}=\frac{v e^{v}+1}{e^{v}} \\ &\Rightarrow y \frac{d v}{d y}=\frac{v e^{v}+1}{e^{v}}-v \\ &\Rightarrow y \frac{d v}{d y}=\frac{v e^{v}+1-v e^{v}}{e^{v}} \end{aligned}

\Rightarrow y \frac{d v}{d y}=\frac{1}{e^{v}} \\

\Rightarrow \int e^{v} d v=\int \frac{d y}{e y} \\                                                                                                \left ( \therefore Integrating\: on\: both\: side \right )

\Rightarrow e^{v}=\log y+c \\

\Rightarrow e^{x} / y=\log y+c                                                                                                                                    \left [ \therefore v=y/x \right ]

Hence this is required solution

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