Question

Solve  ${\displaystyle xy(1+x{y}^2)\frac{dy}{dx}=1}$

Answer 1

${\displaystyle \therefore \frac{dx}{dy}=xy+x^2{y}^3}$

${\displaystyle \therefore \frac{1}{x^2}\frac{dx}{dy}-\frac{1}{x}y=y^3}$  Now, put${\displaystyle -\frac{1}{x}=v}$

${\displaystyle \therefore \frac{dv}{dy}+vy=y^3}$ ...........................${\displaystyle }$

This is linear differential eqn.

∴ Integrating Factor ${\displaystyle =e^{\int ydy}=\frac{y^2}{e^2}}$

The solution of linear diff. eqn is given by,

𝒗.(I.F.) = ∫(𝑰.𝑭.)(𝑹.𝑯.𝑺) + c

${\displaystyle v{e}^{\frac{y^2}{2}}=\int e^{\frac{y^2}{2}}(y^2-2)+c}$

Where c is constant of integration.