Internal problem ID [8270]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 690.
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 x^{3} \sqrt {x^{2}-2 x +1+8 y}}{4 \left (x +1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.373 (sec). Leaf size: 40
dsolve(diff(y(x),x) = 1/4*(-x^2+1+4*x^3*(x^2-2*x+1+8*y(x))^(1/2))/(x+1),y(x), singsol=all)
\[ c_{1}+\frac {4 x^{3}}{3}-2 x^{2}+4 x -4 \ln \left (x +1\right )-\sqrt {x^{2}+8 y \relax (x )-2 x +1} = 0 \]
✓ Solution by Mathematica
Time used: 1.312 (sec). Leaf size: 62
DSolve[y'[x] == (1/4 - x^2/4 + x^3*Sqrt[1 - 2*x + x^2 + 8*y[x]])/(1 + x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{72} \left (4 x^3-6 x^2+15 x+12 \log \left (\frac {1}{x+1}\right )-3-12 c_1\right ) \left (x \left (4 x^2-6 x+9\right )+12 \log \left (\frac {1}{x+1}\right )+3-12 c_1\right ) \\ \end{align*}