Internal problem ID [8312]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 732.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)]]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {-x^{2}-x -a x -a +2 x^{3} \sqrt {x^{2}+2 a x +a^{2}+4 y}}{2 \left (x +1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.484 (sec). Leaf size: 43
dsolve(diff(y(x),x) = 1/2*(-x^2-x-a*x-a+2*x^3*(x^2+2*a*x+a^2+4*y(x))^(1/2))/(x+1),y(x), singsol=all)
\[ c_{1}+\frac {2 x^{3}}{3}-x^{2}-2 \ln \left (x +1\right )+2 x -\sqrt {x^{2}+2 x a +a^{2}+4 y \relax (x )} = 0 \]
✓ Solution by Mathematica
Time used: 1.533 (sec). Leaf size: 54
DSolve[y'[x] == (-1/2*a - x/2 - (a*x)/2 - x^2/2 + x^3*Sqrt[a^2 + 2*a*x + x^2 + 4*y[x]])/(1 + x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{4} \left (-a^2-2 a x-x^2+\frac {1}{9} (x (x (2 x-3)+6)-6 \log (-x-1)-6 c_1){}^2\right ) \\ \end{align*}