Question

Solve the following differential equation:

${\displaystyle (\text{x}+2{\text{y}}^3)\frac{\text{dy}}{\text{dx}}=\text{y}}$

Answer 1

${\displaystyle (\text{x}+2{\text{y}}^3)\frac{\text{dy}}{\text{dx}}=\text{y}}$

∴ ${\displaystyle \frac{\text{x}+{\text{2y}}^3}{\text{y}}=\frac{1}{\left(\frac{\text{dy}}{\text{dx}}\right)}}$

∴ ${\displaystyle \frac{\text{x}}{\text{y}}+2{\text{y}}^2=\frac{\text{dx}}{\text{dy}}}$

∴ ${\displaystyle \frac{\text{dy}}{\text{dx}}-\frac{1}{\text{y}}\cdot \text{x}=2{\text{y}}^2}$    .....(1)

This is the linear differential equation of the form 

${\displaystyle \frac{\text{dx}}{\text{dy}}+\text{P}\cdot \text{x}=\text{Q}}$, where P = ${\displaystyle -\frac{1}{\text{y}}}$ and Q = 2y² 

∴ I.F. = ${\displaystyle {\text{e}}^{\int \text{Pdy}}={\text{e}}^{\int -\frac{1}{\text{y}}\text{dy}}}$

∴ = ${\displaystyle {\text{e}}^{-\log \text{y}}={\text{e}}^{\log \left(\frac{1}{\text{y}}\right)}=\frac{1}{\text{y}}}$

∴ the solution of (1) is given by

∴ ${\displaystyle \text{x}\cdot \left(\text{I.F.}\right)=\int \text{Q}\left(\text{I.F.}\right)\text{dy}+\text{c}}$

∴ ${\displaystyle \text{x}\left(\frac{1}{\text{y}}\right)=\int 2{\text{y}}^2\times \frac{1}{\text{y}}\text{dy}+\text{c}}$

∴ ${\displaystyle \frac{\text{x}}{\text{y}}=2\int \text{y}\text{dx}+\text{c}}$

∴ ${\displaystyle \frac{\text{x}}{\text{y}}=2\cdot \frac{{\text{y}}^2}{2}+\text{c}}$

∴ x = y(c + y²)

This is the general solution.