9. differential equations — ગણિત ધોરણ 12 સાયન્સ — Question

$\Rightarrow \quad y \frac{d x}{d y}=x+2 y^{3}$

$\Rightarrow \quad \frac{d x}{d y}=\frac{x}{y}+2 y^{2}$

$\Rightarrow \quad \frac{d x}{d y}-\frac{x}{y}+2 y^{2}$

This is a linear differential cquation. On comparing it with $\frac{d x}{d y}+P x=Q,$ we get $P=-\frac{1}{y}, Q=2 y^{2}$

I.F. $=e^{\int-\frac{1}{y} d y}=e^{-\log y}=\frac{1}{y}$

The general solution is:

$x \frac{1}{y}=\int 2 y^{2} \cdot \frac{1}{y} d y+C$

$\Rightarrow \quad \frac{x}{y}=\int 2 y d y+C$

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

$\Rightarrow \quad x=y^{3}+C y$