Question

What is the solution of the differential equation $\frac{dx}{dy}+\frac{x}{y}-y^2=0$?
where $c$ is an arbitrary constant.

• A. $xy=x^4+c$
• B. $xy=y^4+c$
• C. $4xy=y^4+c$
• D. $3xy=y^3+c$

Answer 1

Answer: (C)

$4xy=y^4+c$

$\frac{dx}{dy}+\frac{x}{y}-y^2=0\Rightarrow \frac{dx}{dy}=\frac{y^3-x}{y}\Rightarrow ydx=y^{3}dy-xdy\Rightarrow xdy+ydx=y^{3}dy\Rightarrow d\left(xy\right)=d\left(\frac{y^4}{4}\right)$

Integrating both the sides, we get

$y^4+c=4xy$