Internal problem ID [8286]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 706.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [y=_G(x,y')]
Solve \begin {gather*} \boxed {y^{\prime }+\frac {\left (-\ln \left (y-1\right )+\ln \left (1+y\right )+2 \ln \relax (x )\right ) x \left (1+y\right )^{2}}{8}=0} \end {gather*}
✓ Solution by Maple
Time used: 1.343 (sec). Leaf size: 101
dsolve(diff(y(x),x) = -1/8*(-ln(-1+y(x))+ln(y(x)+1)+2*ln(x))*x*(y(x)+1)^2,y(x), singsol=all)
\begin{align*} y \relax (x ) = {\mathrm e}^{\RootOf \left (-x^{2} {\mathrm e}^{\textit {\_Z}} \ln \left (\frac {{\mathrm e}^{\textit {\_Z}}-2}{x^{2}}\right )+\textit {\_Z} \,x^{2} {\mathrm e}^{\textit {\_Z}}+8 \,{\mathrm e}^{\textit {\_Z}}-16\right )}-1 \\ \int _{\textit {\_b}}^{y \relax (x )}\frac {1}{2 \left (-\frac {x^{2} \left (\textit {\_a} +1\right ) \ln \left (-1+\textit {\_a} \right )}{2}+\frac {x^{2} \left (\textit {\_a} +1\right ) \ln \left (\textit {\_a} +1\right )}{2}+x^{2} \left (\textit {\_a} +1\right ) \ln \relax (x )+4 \textit {\_a} -4\right ) \left (\textit {\_a} +1\right )}d \textit {\_a} +\frac {\ln \relax (x )}{8}-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 23.709 (sec). Leaf size: 610
DSolve[y'[x] == -1/8*(x*(2*Log[x] - Log[-1 + y[x]] + Log[1 + y[x]])*(1 + y[x])^2),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {-2 \log (x) x^2+\log (K[2]-1) x^2-\log (K[2]+1) x^2-8}{2 \left (2 \log (x) x^2-\log (K[2]-1) x^2+\log (K[2]+1) x^2+K[2] \left (2 \log (x) x^2-\log (K[2]-1) x^2+\log (K[2]+1) x^2+8\right )-8\right )}-\int _1^x\left (-\frac {K[1] (K[2]+1) \left (\frac {1}{K[2]+1}-\frac {1}{K[2]-1}\right )}{2 K[2] \log (K[1]) K[1]^2+2 \log (K[1]) K[1]^2-K[2] \log (K[2]-1) K[1]^2-\log (K[2]-1) K[1]^2+K[2] \log (K[2]+1) K[1]^2+\log (K[2]+1) K[1]^2+8 K[2]-8}-\frac {K[1] (2 \log (K[1])-\log (K[2]-1)+\log (K[2]+1))}{2 K[2] \log (K[1]) K[1]^2+2 \log (K[1]) K[1]^2-K[2] \log (K[2]-1) K[1]^2-\log (K[2]-1) K[1]^2+K[2] \log (K[2]+1) K[1]^2+\log (K[2]+1) K[1]^2+8 K[2]-8}+\frac {K[1] (K[2]+1) (2 \log (K[1])-\log (K[2]-1)+\log (K[2]+1)) \left (-\frac {K[2] K[1]^2}{K[2]-1}+2 \log (K[1]) K[1]^2-\log (K[2]-1) K[1]^2+\log (K[2]+1) K[1]^2-\frac {K[1]^2}{K[2]-1}+\frac {K[2] K[1]^2}{K[2]+1}+\frac {K[1]^2}{K[2]+1}+8\right )}{\left (2 K[2] \log (K[1]) K[1]^2+2 \log (K[1]) K[1]^2-K[2] \log (K[2]-1) K[1]^2-\log (K[2]-1) K[1]^2+K[2] \log (K[2]+1) K[1]^2+\log (K[2]+1) K[1]^2+8 K[2]-8\right )^2}\right )dK[1]+\frac {1}{2 (K[2]+1)}\right )dK[2]+\int _1^x-\frac {K[1] (2 \log (K[1])-\log (y(x)-1)+\log (y(x)+1)) (y(x)+1)}{2 \log (K[1]) K[1]^2-\log (y(x)-1) K[1]^2+\log (y(x)+1) K[1]^2+2 \log (K[1]) y(x) K[1]^2-\log (y(x)-1) y(x) K[1]^2+\log (y(x)+1) y(x) K[1]^2+8 y(x)-8}dK[1]=c_1,y(x)\right ] \]