Internal problem ID [9109]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 774.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class A`]]
\[ \boxed {y^{\prime }-\frac {-4 y x -x^{3}-2 a \,x^{2}-4 x +8}{8 y+2 x^{2}+4 a x +8}=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 50
dsolve(diff(y(x),x) = (-4*x*y(x)-x^3-2*a*x^2-4*x+8)/(8*y(x)+2*x^2+4*a*x+8),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {2 a^{2} x +a \,x^{2}+8 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} x}{4}} {\mathrm e}^{-\frac {a}{2}} {\mathrm e}^{-1} {\mathrm e}^{\frac {c_{1} a^{2}}{4}}}{2}\right )+4 a +8}{4 a} \]
✓ Solution by Mathematica
Time used: 5.106 (sec). Leaf size: 72
DSolve[y'[x] == (8 - 4*x - 2*a*x^2 - x^3 - 4*x*y[x])/(8 + 4*a*x + 2*x^2 + 8*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {8 W\left (-e^{-\frac {a^2 x}{4}-1+c_1}\right )+2 a^2 x+a \left (x^2+4\right )+8}{4 a} y(x)\to -\frac {2 a^2 x+a \left (x^2+4\right )+8}{4 a} \end{align*}