Question

ques

Answer 1

Solve
\[(y^2+2x^2y)dx+(2x^3-xy)dy=0\]
\[y(y+2x^2)dx+x(2x^2-y)dy=0\]
Thus we have
\[M=yf(x,y) \space \space \space ; \space \space \space N=xg(x,y)\]
In this case,
\[I.F.=\frac{1}{Mx-Ny}\]\[I.F.=\frac{1}{xy^2+2x^3y-2x^3y+xy^2}=\frac{1}{2xy^2}\]
Multiplying our original equation with IF we get
\[(\frac{1}{2x}+\frac{x}{y})dx+(\frac{x^2}{y^2}-\frac{1}{2y})dy=0\]
But if we calculate the partial derivatives..
\[\frac{\partial M}{\partial y}=-\frac{x}{y^2} \space \space \space ; \space \space \space \frac{\partial N}{\partial x}=\frac{2x}{y^2}\]
They are not coming to be equal why??

Answer 2

Ok, I've understood the problem now, it should be
\[f(xy)\] and \[g(xy)\]
and not
\[f(x,y)\]\[g(x,y)\]
Should be a function of the product of xy