Internal problem ID [8449]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 871.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]
\[ \boxed {y^{\prime }-\frac {2 x y^{2}+4 y \ln \left (2 x +1\right ) x +2 \ln \left (2 x +1\right )^{2} x +y^{2}-2+\ln \left (2 x +1\right )^{2}+2 y \ln \left (2 x +1\right )}{2 x +1}=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 66
dsolve(diff(y(x),x) = 1/(2*x+1)*(2*x*y(x)^2+4*y(x)*ln(2*x+1)*x+2*ln(2*x+1)^2*x+y(x)^2-2+ln(2*x+1)^2+2*y(x)*ln(2*x+1)),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\frac {4 x \ln \left (2 x +1\right )}{2 x +1}+\frac {2 \ln \left (2 x +1\right )}{2 x +1}}{2 \left (\frac {2 x}{2 x +1}+\frac {1}{2 x +1}\right )}+\frac {1}{-x +c_{1}} \]
✓ Solution by Mathematica
Time used: 0.361 (sec). Leaf size: 34
DSolve[y'[x] == (-2 + Log[1 + 2*x]^2 + 2*x*Log[1 + 2*x]^2 + 2*Log[1 + 2*x]*y[x] + 4*x*Log[1 + 2*x]*y[x] + y[x]^2 + 2*x*y[x]^2)/(1 + 2*x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\log (2 x+1)+\frac {1}{-x+c_1} \\ y(x)\to -\log (2 x+1) \\ \end{align*}