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Q.
The equation of curve passing through $(3,4)$ and satisfying the differential equation $y\left(\frac{d y}{d x}\right)^2+(x-y) \frac{d y}{d x}-x=0$ is
Differential Equations
Solution:
$y\left(\frac{d y}{d x}\right)^2+x \frac{d y}{d x}-y \frac{d y}{d x}-x=0$
$y \frac{d y}{d x}\left(\frac{d y}{d x}-1\right)+x\left(\frac{d y}{d x}-1\right)=0$
$\left(y \frac{d y}{d x}+x\right)\left(\frac{d y}{d x}-1\right)=0$
$\therefore $ either $y d y+x d x=0$ or $ d y-d x=0$
since the curves pass through the point $(3,4)$
$\therefore x^2+y^2=25 $ or $ x-y+1=0$