Question

The solution of the differential equation $\left(x^2-y{x}^2\right)\frac{dy}{dx}+y^2+x{y}^2=0$ is

• A. $\log \left(\frac{x}{y}\right)=\frac{1}{x}+\frac{1}{y}+c$
• B. $\log \left(\frac{y}{x}\right)=\frac{1}{x}+\frac{1}{y}+c$
• C. $\log \left(xy\right)=\frac{1}{x}+\frac{1}{y}+c$
• D. $\log \left(xy\right)+\frac{1}{x}+\frac{1}{y}=c$

Answer 1

The correct option is A $\log \left(\frac{x}{y}\right)=\frac{1}{x}+\frac{1}{y}+c$

$(x^2-y{x}^2)\frac{dy}{dx}+y^2+x{y}^2=0$

$x^{2}dy-y{x}^{2}dy+y^{2}dx+x{y}^{2}dx=0$

$x^{2}dy+y^{2}dx=xy(xdy-ydx)$

$\frac{dy}{y^2}+\frac{dx}{x^2}=\frac{x}{y}\frac{d(\frac{y}{x})}{dx}$

Integrating on both sides

$-\frac{1}{y}-\frac{1}{x}=\ln \frac{y}{x}$

$\ln (\frac{x}{y})=\frac{1}{x}+\frac{1}{y}+c$