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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.

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

Answers (1)

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}

So,v+x\frac{dv}{dx}=\frac{3vx^{2}+v^{2}x^{2}}{3x^{2}}

\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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