Internal problem ID [8183]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 603.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(y)]]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {2 F \left (y+\ln \left (2 x +1\right )\right ) x +F \left (y+\ln \left (2 x +1\right )\right )-2}{2 x +1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.019 (sec). Leaf size: 42
dsolve(diff(y(x),x) = 1/(2*x+1)*(2*F(y(x)+ln(2*x+1))*x+F(y(x)+ln(2*x+1))-2),y(x), singsol=all)
\begin{align*} y \relax (x ) = -\ln \left (2 x +1\right )+\RootOf \left (F \left (\textit {\_Z} \right )\right ) \\ y \relax (x ) = -\ln \left (2 x +1\right )+\RootOf \left (-x +\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} +c_{1}\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.28 (sec). Leaf size: 117
DSolve[y'[x] == (-2 + F[Log[1 + 2*x] + y[x]] + 2*x*F[Log[1 + 2*x] + y[x]])/(1 + 2*x),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}-\frac {F(K[2]+\log (2 x+1)) \int _1^x-\frac {2 F'(K[2]+\log (2 K[1]+1))}{F(K[2]+\log (2 K[1]+1))^2 (2 K[1]+1)}dK[1]-1}{F(K[2]+\log (2 x+1))}dK[2]+\int _1^x\left (\frac {2}{F(\log (2 K[1]+1)+y(x)) (2 K[1]+1)}-1\right )dK[1]=c_1,y(x)\right ] \]