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Provide solution for RD Sharma maths class 12 chapter Differential Equation exercise 21.3 question 14

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Answer:

y=-x-1  is a solution of differential equation

Hint:

Differentiate the solution and modify the given differential equation

Given:

y=-x-1

Solution:

Differentiating on both sides with respect to x

\begin{aligned} &\frac{d y}{d x}=-\frac{d(x)}{d x}-\frac{d(1)}{d x} \\\\ &\frac{d y}{d x}=-1 \end{aligned}            ...............(i)

Now the given equation has to be modified

\begin{aligned} &(y-x) d y-\left(y^{2}-x^{2}\right) d x=0 \\\\ &(y-x) d y=\left(y^{2}-x^{2}\right) d x \\\\ &\frac{d y}{d x}=\frac{y-x}{y^{2}-x^{2}} \end{aligned}

\begin{aligned} &\frac{d y}{d x}=\frac{y-x}{(y-x)(y+x)} \\\\ &\frac{d y}{d x}=\frac{1}{y+x} \end{aligned}            ..............(ii)

Comparing equation (i) and (ii)

\begin{aligned} &\frac{1}{y+x}=-1 \\\\ &1=-(y+x) \\\\ &y=-x-1 \end{aligned}

Thus,it is proved that y=-x-1 is a solution of differential equation.

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