Internal problem ID [8274]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 694.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {x +1+2 \sqrt {4 x^{2} y+1}\, x^{3}}{2 x^{3} \left (x +1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.449 (sec). Leaf size: 30
dsolve(diff(y(x),x) = 1/2*(x+1+2*(4*x^2*y(x)+1)^(1/2)*x^3)/x^3/(x+1),y(x), singsol=all)
\[ -2 \ln \left (x +1\right )-\frac {\sqrt {4 x^{2} y \relax (x )+1}}{x}+2 x +c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 1.233 (sec). Leaf size: 49
DSolve[y'[x] == (1/2 + x/2 + x^3*Sqrt[1 + 4*x^2*y[x]])/(x^3*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x^2-\frac {1}{4 x^2}-2 c_1 x+\frac {1}{4} \log \left ((x+1)^2\right ) \left (-4 x+\log \left ((x+1)^2\right )+4 c_1\right )+c_1{}^2 \\ \end{align*}