Question

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

Answer 1

`(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`