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Solve X Y ( 1 + X Y 2 ) D Y D X = 1 - Applied Mathematics 2

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Sum

Solve  `xy(1+xy^2)(dy)/(dx)=1`

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Solution

`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.

Concept: Linear Differential Equations
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