Find the general solution for the differential equation (x^2 y-x^… - Vidyadip

Question

Find the general solution for the differential equation $\left(x^2 y-x^2\right) d x+\left(x y^2-y^2\right) d y=0$

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Answer

The given differential equation is,
$x^2(y-1) d x+y^2(x-1) d y=0$
$\frac{x^2}{x-1} d x+\frac{y^2}{y-1} d y=0$
Add and subtract $1$ in numerators we have,
$\frac{x^2-1+1}{(x-1)} d x+\frac{y^2-1+1}{(y-1)} d y=0$
By the identity $\left(a^2-b^2\right)=(a+b) \cdot(a-b)$
$\frac{(x+1)(x-1)+1}{(x-1)} d x+\frac{(y+1)(y-1)+1}{(y-1)} d y=0$
Splitting the terms,
$(x+1) d x+\frac{1}{(x-1)} d x+(y+1) dy +\frac{1}{(y-1)} d y=0$
Integrating, we get,
$\int(x+1) d x+\int \frac{1}{(x-1)} d x+\int(y+1) d y+\int \frac{1}{(y-1)} d y=C$
$\frac{x^2}{2}+x+\log |x-1|+\frac{y^2}{2}+y+\log |y-1|= C$
$\frac{1}{2} \cdot\left(x^2+y^2\right)+(x+y)+\log |(x-1)(y-1)|= C $
This is the required solution

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