Internal problem ID [8335]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 757.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class A`]]
\[ \boxed {y^{\prime }-\frac {-4 y x +x^{3}+2 x^{2}-4 x -8}{-8 y+2 x^{2}+4 x -8}=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 26
dsolve(diff(y(x),x) = (-4*x*y(x)+x^3+2*x^2-4*x-8)/(-8*y(x)+2*x^2+4*x-8),y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{2}}{4}+2 \operatorname {LambertW}\left (\frac {c_{1} {\mathrm e}^{-\frac {x}{4}} {\mathrm e}^{-\frac {1}{2}}}{2}\right )+\frac {x}{2}+1 \]
✓ Solution by Mathematica
Time used: 3.974 (sec). Leaf size: 45
DSolve[y'[x] == (-8 - 4*x + 2*x^2 + x^3 - 4*x*y[x])/(-8 + 4*x + 2*x^2 - 8*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 2 W\left (-e^{-\frac {x}{4}-1+c_1}\right )+\frac {1}{4} x (x+2)+1 \\ y(x)\to \frac {1}{4} x (x+2)+1 \\ \end{align*}