Internal problem ID [8292]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 712.
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}+2 x +1+2 x^{3} \sqrt {x^{2}+2 x +1-4 y}}{2 \left (x +1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.407 (sec). Leaf size: 38
dsolve(diff(y(x),x) = 1/2*(x^2+2*x+1+2*x^3*(x^2+2*x+1-4*y(x))^(1/2))/(x+1),y(x), singsol=all)
\[ c_{1}-\frac {2 x^{3}}{3}+x^{2}-2 x +2 \ln \left (x +1\right )-\sqrt {x^{2}-4 y \relax (x )+2 x +1} = 0 \]
✓ Solution by Mathematica
Time used: 1.219 (sec). Leaf size: 47
DSolve[y'[x] == (1/2 + x + x^2/2 + x^3*Sqrt[1 + 2*x + x^2 - 4*y[x]])/(1 + x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{4} \left (x^2+2 x-\frac {1}{9} \left (x (x (2 x-3)+6)+6 \log \left (\frac {1}{x+1}\right )-6 c_1\right ){}^2+1\right ) \\ \end{align*}