# Solve ( Y − X Y 2 ) D X − ( X + X 2 Y ) D Y = 0 - Applied Mathematics 2

Sum

Solve (y-xy^2)dx-(x+x^2y)dy=0

#### Solution

(y-xy^2)dx-(x+x^2y)dy=0  ---------------------(1)

Comparing the given eqn with M dx +N dy = 0

thereforeM=(y-xy^2)   thereforeN=(x+x^2y)

(delM)/(dely)=1-2xy       (delN)/(delx)=-(1+2xy)

(delM)/(dely)!=(delN)/(delx)

The given differential eqn is not exact diff. eqn.
But the given diff . eqn is in the form of 𝒚.𝒇(𝒙𝒚)𝒅𝒙+𝒙𝒇(𝒙𝒚)𝒅𝒚=𝟎

Integrating factor = I.F. =1/(Mx-Ny)=1/(xy-x^2y^2+xy+x^2y^2)=1/(2xy)

Multiply the I.F. to eqn (1)

(1/(2x)-y/2)dx-(1/(2y)+x/2)dy=0

thereforeM_1=(1/(2x)-y/2)              N_1=-(1/(2y)+x/2)

intM_1dx=int(1/(2x)-y/2)dx=1/2(logx)-(xy)/2

del/(dely)intM_1 dx=(-x)/2

int[N_1-del/(dely)intM_1 dx]dy=int(-1)/(2y)dy=(-1)/2(logy)

The solution of given diff. eqn is given by,

intM_1dx+int[N_1-del/(dely)intM_1dx]dy=c

therefore1/2(logx)-(xy)/2-1/2(logy)=c

thereforelog(sqrtx/sqrty)-(xy)/2=c

Concept: Equations Reducible to Exact Form by Using Integrating Factors
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