Internal problem ID [9025]
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)]`]]
\[ \boxed {y^{\prime }-\frac {-x^{2}+1+4 x^{3} \sqrt {x^{2}-2 x +1+8 y}}{4 \left (1+x \right )}=0} \]
✓ Solution by Maple
Time used: 0.125 (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)
\[ c_{1} +\frac {4 x^{3}}{3}-2 x^{2}+4 x -4 \ln \left (x +1\right )-\sqrt {x^{2}-2 x +1+8 y \left (x \right )} = 0 \]
✓ Solution by Mathematica
Time used: 1.35 (sec). Leaf size: 108
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]
\[ y(x)\to \frac {2 x^6}{9}-\frac {2 x^5}{3}+\frac {11 x^4}{6}-\frac {2}{3} (3+2 c_1) x^3+\left (\frac {15}{8}+2 c_1\right ) x^2+\left (\frac {4 x^3}{3}-2 x^2+4 x-4 c_1\right ) \log \left (\frac {1}{x+1}\right )+2 \log ^2\left (\frac {1}{x+1}\right )+\left (\frac {1}{4}-4 c_1\right ) x-\frac {1}{8}+2 c_1{}^2 \]