Question

Solve the differential equation $xdy+2ydx=0$ when x=2, y=1.

Answer 1

$xdy+2ydx=0$

$xdy=-2ydx$

$\frac{dy}{-2y}=\frac{dx}{x}$

Integrating both the sides

$\int \frac{-dy}{2y}=\int \frac{dx}{x}$

$\frac{-1}{2}logy=logx+loge$

$log{y}^{-1/2}=logxc$

taking anti log

$\frac{1}{\sqrt{y}}=xc$

at $x=2,y=1\Rightarrow c=\frac{1}{2}$

$\therefore x\sqrt{y}=2\Rightarrow x^{2}y=4$