Question

Solve  `xy(1+xy^2)(dy)/(dx)=1`

Answer 1

`therefore(dx)/(dy)=xy+x^2y^3`

`therefore 1/x^2(dx)/(dy)-1/xy=y^3`  Now, put`-1/x=v`

`therefore (dv)/(dy)+vy=y^3` ...........................``

This is linear differential eqn.

∴ Integrating Factor `=e^(intydy)=y^2/e^2`

The solution of linear diff. eqn is given by,

𝒗.(I.F.) = ∫(𝑰.𝑭.)(𝑹.𝑯.𝑺) + c

`ve^(y^2/2)=inte^(y^2/2)(y^2-2)+c`

Where c is constant of integration.