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Provide Solution For  R. D. Sharma Maths Class 12 Chapter 21 Differential Equations Exercise 21.9 Question 14 Maths Textbook Solution.

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Answer: \frac{-3x}{y}=log|x|+c

Given: 3x^{2}dy=\left ( 3xy+y^{2} \right )dx

To solve: we have to solve given differential equation

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

Solution: we have,

3x^{2}dy=\left ( 3xy+y^{2} \right )dx

\Rightarrow \frac{dy}{dx}=\frac{3xy+y^{2}}{3x^{2}}

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


\begin{aligned} &\Rightarrow x \frac{d v}{d y}=\frac{3 v+v^{2}}{3}-v \\ &\Rightarrow x \frac{d v}{d y}=\frac{v^{2}}{3} \\ &\Rightarrow 3\left(\frac{-1}{v}\right)=\log |x|+c \; \; \; \; \; \; \quad\left[\therefore \int \frac{x^{2}+1}{3+1}+c\right] \\ &\Rightarrow \frac{-3 x}{y}=\log |x|+c \; \; \; \; \; \quad[\therefore v=y / x] \end{aligned}

This is required solution

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