Mathematics - Differential Equation Question with Solution | TestHub

Given differential equation,

$ydx+x{y}^{2}dx=xdy$

$\Rightarrow$$\frac{xdy-ydx}{y^2}=xdx$

$\Rightarrow$$-d\left(\frac{x}{y}\right)=d\left(\frac{x^2}{2}\right)$

On integrating we get,

$\Rightarrow -\frac{x}{y}=\frac{x^2}{2}+C$

$\because$It passes through$(1,-1)$.

$\therefore$$1=\frac{1}{2}+C \Rightarrow C=\frac{1}{2}$

$\therefore$$x^2+1+\frac{2x}{y}=0 \Rightarrow y=\frac{-2x}{x^2+1}$

$\therefore$$f\left(-\frac{1}{2}\right)=\frac{4}{5}$