Internal problem ID [8351]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 771.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _rational, [_Abel, 2nd type, class A]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {-4 a y x -a^{2} x^{3}-2 x^{2} b a -4 a x +8}{8 y+2 a \,x^{2}+4 x b +8}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.141 (sec). Leaf size: 84
dsolve(diff(y(x),x) = (-4*y(x)*a*x-a^2*x^3-2*a*x^2*b-4*a*x+8)/(8*y(x)+2*a*x^2+4*b*x+8),y(x), singsol=all)
\[ y \relax (x ) = \frac {-a \,x^{2} b -2 b^{2} x -4 b +4 \,{\mathrm e}^{-\frac {a \,b^{2} x +2 c_{1} b^{2}+4 \LambertW \left (-\frac {{\mathrm e}^{-\frac {b^{2} x}{4}} {\mathrm e}^{-\frac {c_{1} b^{2}}{2 a}} {\mathrm e}^{-\frac {b}{2}} {\mathrm e}^{-1}}{2}\right ) a +2 b a +4 a}{4 a}}-8}{4 b} \]
✓ Solution by Mathematica
Time used: 60.288 (sec). Leaf size: 46
DSolve[y'[x] == (8 - 4*a*x - 2*a*b*x^2 - a^2*x^3 - 4*a*x*y[x])/(8 + 4*b*x + 2*a*x^2 + 8*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {b \left (a x^2+2 b x+4\right )+8 \text {ProductLog}\left (-e^{-\frac {b^2 x}{4}-1+c_1}\right )+8}{4 b} \\ \end{align*}