Obtain the complete primitive and singular solutions of the differential equation: $$y-x\frac{dy}{dx}-(\frac{dy}{dx})^2=0$$
Could someone please explain to me how the equation can be solved?
Does it need to be translated to $$y''+xy'-y=0$$
and which type of method do you need to use to solve it?
Let us rewrite the equation as $$y-xy'-(y')^2=0$$ Differentiating both sides with respect to $x$, we get: $$ y'-y'-xy''-2y'y''=0$$ or equivalently: $$ xy''+2y'y''=y''(x+2y')=0$$ Thus $y''=0$ or $x+2y'=0$. If $y''=0$, then $y=c_1x+c_2$. If $x+2y'=0$, then $y'=\frac{-x}{2}$ and $y=-\frac{x^2}{4}+c_3$.
Therefore, the solution to your differential equation is $y=-\frac{x^2}{4}+c_3$ or $y=c_1x+c_2$.