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

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

Answers (1)

Answer: x^{2}y-xy^{2}=C

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

To find: 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

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

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

It is homogeneous differential equation.

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

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

\begin{aligned} &\Rightarrow v+x \frac{d v}{d x}=\frac{v^{2}-2 v}{1-2 v} \\ &\Rightarrow x \frac{d v}{d x}=\frac{v^{2}-2 v}{1-2 v}-v \\ &\Rightarrow x \frac{d v}{d x}=\frac{v^{2}-2 v-v+2 v^{2}}{1-2 v} \\ &\Rightarrow x \frac{d v}{d x}=\frac{3 v^{2}-3 v}{1-2 v} \end{aligned}

\Rightarrow \frac{1-2 v}{3\left(v^{2}-v\right)} d v=\frac{d v}{x} \\

\Rightarrow \frac{-(2 v-1)}{3\left(v^{2}-v\right)} d v=\frac{d x}{d x} \\

\Rightarrow \int \frac{2 v-1)}{\left.v^{2}-v\right)} d v=-3 \int \frac{d x}{x} \\                [Integrating \; \; on \: \: both\: \: side]

\Rightarrow \log \left|v^{2}-v\right|=-3 \log |x|+\log c . \\

\Rightarrow v^{2}-v=\frac{c}{x^{3}} \\

\Rightarrow \frac{y^{2}}{x^{2}}-\frac{y}{x}=\frac{c}{x^{3}} \\

\Rightarrow \frac{y^{2}-x y}{x^{2}}=\frac{c}{x^{3}} \\

\Rightarrow y^{2}-x y=\frac{c}{x} \\

\Rightarrow x y^{2}-x^{2} y=c \\

                                                         

                                                      

Posted by

infoexpert21

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