Show if equation $M(x,y)dx+N(x,y)dy=0$ is homogeneous & $M(x,y)x+N(x,y)y\neq0$, then $[M(x,y)x+N(x,y)y]^{-1}$ is integrating factor of equation.

Question

Questions

• Show that if the equation $$M(x,y)dx+N(x,y)dy=0 \qquad(A)$$ is homogeneous and $M(x,y)\cdot x+N(x,y)\cdot y\neq0$, then $[M(x,y)x+N(x,y)y]^{-1}$ is an integrating factor of $(A)\\$

• Show that if the equation $M(x,y)dx+N(x,y)dy=0$ is both homogeneous and exact and if $M(x,y)x+N(x,y)y$ is not a constant, then the solution of this equation is $M(x,y)x+N(x,y)y=c$, where $c$ is an arbitrary constant.

[Problems taken from Differential Equations by Shepley L. Ross, Page $68$, Exercise-$18$ and $19$]

Any Hint appreciated.