Question

The solution of differential equation $(x+2y^2)\frac{dy}{dx}=y$ is:

• A. $x=2y{x}^{-1}+c$
• B. $x{y}^{-1}=2y+c$
• C. $xy=c$
• D. $x+y=c{x}^{-1}$

Answer 1

The correct option is B $x{y}^{-1}=2y+c$

$(x+2y^2)\frac{dy}{dx}=y$
$\Rightarrow y\frac{dx}{dy}=x+2y^2$
$\Rightarrow \frac{dx}{dy}-\frac{x}{y}=2y$
which is a linear differential equation with x as dependent variable.
Here, $P=-\frac{1}{y},Q=2y$
Integrating factor $I.F.=e^{\int Pdy}$
$=e^{\int \frac{-1}{y}dy}=e^{-\log y}=\frac{1}{y}$
$I.F=\frac{1}{y}$
Solution of given differential eqn is
$x\left(\frac{1}{y}\right)=\int 2y\left(\frac{1}{y}\right)dy+c$
$\Rightarrow \frac{x}{y}=2y+c$
$\Rightarrow x{y}^{-1}=2y+c$