Question

Find the general solution of $(x+2y^3)\frac{dy}{dx}=y$

Answer 1

Given:
$(x+2y^3)\frac{dy}{dx}=y$
$\Rightarrow \frac{dy}{dx}=\frac{y}{x+2y^3}$
$\Rightarrow \frac{dx}{dy}=\frac{x+2y^3}{y}$
$\Rightarrow \frac{dx}{dy}=\frac{x}{y}+2y^2$
$\Rightarrow \frac{dx}{dy}-\frac{x}{y}=2y^2$
Compare with standard form of linear differential equation
$\therefore \text{Standard form :}\frac{dx}{dy}+Px=Q$
$P=\frac{-1}{y}$,$Q=2y^2$
$I.F.=e^{\int \frac{-1}{y}dy}$ $(\because I.F.=e^{\int P.dy})$
$=e^{-\ln |y|-\frac{1}{|y|}}$
So, the solution of the equation is
$x\cdot \frac{1}{|y|}=\int 2y^2\cdot \frac{1}{|y|}dy+c$
$[\because x(I.F.)=\int Q.(I.F.)dy+c]$
$\Rightarrow \pm \frac{x}{y}=\pm \int 2ydy+c$ $[\because |y|=\pm y]$
$\Rightarrow \frac{x}{y}=\int 2ydy\pm c$
$\Rightarrow \frac{x}{y}=2\cdot \frac{y^2}{2}+k$
$\Rightarrow x=y(y^2+k)$
Hence, the required solution is
$x=y(y^2+k)$