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Solve the Ode ( Y + 1 3 Y 3 + 1 2 X 2 ) D X + ( X + X Y 2 ) D Y = 0 - Applied Mathematics 2

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Sum

Solve the ODE `(y+1/3y^3+1/2x^2)dx+(x+xy^2)dy=0`

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Solution

Compare the given diff. eqn with Mdx + Ndy = 0

`therefore M = (y+1/3y^3+1/2x^2)`   `thereforeN=(x+xy^2)`

`(delM)/(dely)=1+y^2`                         `(delN)/(dely)=1+y^2`

`therefore (delM)/(dely)=(delN)/(dely)`

The given differential eqn is exact.
The solution of exact differential eqn is given by,

`intMdx+int[N-del/(dely)Mdx]dy=c`

`intMdx=int (y+1/3y^3+1/2x^2)dx=xy+x/3y^3+x^3/6`

`del/(dely)intMdx=x+xy^2`

`int[N-del/(dely)Mdx]dy=int[x+xy^2-(x+xy^2)]dy=0`

`therefore xy+x/3y^3+x^3/6=c`

Concept: Exact Differential Equations
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