Internal problem ID [8238]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 658.
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}-1-4 \sqrt {x^{2}-2 x +1+8 y}}{4 x +4}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.348 (sec). Leaf size: 28
dsolve(diff(y(x),x) = -1/4*(x^2-1-4*(x^2-2*x+1+8*y(x))^(1/2))/(x+1),y(x), singsol=all)
\[ c_{1}+4 \ln \left (x +1\right )-\frac {1}{4}-\sqrt {x^{2}+8 y \relax (x )-2 x +1} = 0 \]
✓ Solution by Mathematica
Time used: 1.16 (sec). Leaf size: 39
DSolve[y'[x] == (1/4 - x^2/4 + Sqrt[1 - 2*x + x^2 + 8*y[x]])/(1 + x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{8} \left (x-4 \log \left (\frac {1}{x+1}\right )-1-4 c_1\right ) \left (x+4 \log \left (\frac {1}{x+1}\right )-1+4 c_1\right ) \\ \end{align*}