Question

Find the differential equations of the following equations by integrating factors by inspection. Show complete solution.

xdy - ydx = x^4 ydy + x^3 y^2dx 

Answer 1

$\displaystyle xdy -ydx = x^4ydy + x^3y^2dx\\ =xdy -ydx - x^4ydy-x^3y^2dx=0\\ =xdy - ydx - x^3y(xdy+ydx)=0\\ \text{Divide through by $x^2$ to obtain}\\ \frac{xdy - ydx}{x^2}-xyd(xy)=0\\ =d(\frac{y}{x})-xyd(xy)=0 \qquad (1)\\ \text{Integrating (1) we have}\\ \frac{y}{x}-\frac{1}{2}(xy)^2=c$