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

Sum

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

#### 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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