Problem $2.$ For a first order ODE in the following form:

$M(x,y)dx+N(x,y)dy=0$

Find the condition for $N$ and $M$ such that the following functional form of integrating factors exists

$(x,y)=(x{y}^2)$

Ans:

Check whether this is true: $\frac{(\frac{\text{d}\text{e}\text{l}\text{M}}{\text{d}\text{e}\text{l}\text{y}}-\frac{\text{d}\text{e}\text{l}\text{N}}{\text{d}\text{e}\text{l}\text{x}})}{N{y}^2-2xyM}=(),$ where $=x{y}^2$