Internal problem ID [8426]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 846.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [y=_G(x,y')]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {1}{-x +\left (\frac {1}{y}+1\right ) x +f_{1}\left (\left (\frac {1}{y}+1\right ) x \right ) x^{2}-f_{1}\left (\left (\frac {1}{y}+1\right ) x \right ) x^{2} \left (\frac {1}{y}+1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.035 (sec). Leaf size: 71
dsolve(diff(y(x),x) = 1/(-x+(1/y(x)+1)*x+_F1((1/y(x)+1)*x)*x^2-_F1((1/y(x)+1)*x)*x^2*(1/y(x)+1)),y(x), singsol=all)
\begin{align*} y \relax (x ) = \RootOf \left (f_{1}\left (\frac {\left (\textit {\_Z} +1\right ) x}{\textit {\_Z}}\right ) x \textit {\_Z} +f_{1}\left (\frac {\left (\textit {\_Z} +1\right ) x}{\textit {\_Z}}\right ) x -\textit {\_Z} \right ) \\ y \relax (x ) = {\mathrm e}^{\RootOf \left (-\textit {\_Z} -\left (\int _{}^{\frac {x \,{\mathrm e}^{\textit {\_Z}}}{{\mathrm e}^{\textit {\_Z}}-1}}\frac {1}{\textit {\_a} \left (f_{1}\left (\textit {\_a} \right ) \textit {\_a} -1\right )}d \textit {\_a} \right )+c_{1}\right )}-1 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.566 (sec). Leaf size: 346
DSolve[y'[x] == (-x + x^2*F1[x*(1 + y[x]^(-1))] + x*(1 + y[x]^(-1)) - x^2*F1[x*(1 + y[x]^(-1))]*(1 + y[x]^(-1)))^(-1),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {x \text {F1}\left (x \left (1+\frac {1}{K[2]}\right )\right )-1}{x \text {F1}\left (x \left (1+\frac {1}{K[2]}\right )\right )+x K[2] \text {F1}\left (x \left (1+\frac {1}{K[2]}\right )\right )-K[2]}-\int _1^x\left (\frac {\text {F1}\left (K[1] \left (1+\frac {1}{K[2]}\right )\right )-\frac {K[1] \text {F1}'\left (K[1] \left (1+\frac {1}{K[2]}\right )\right )}{K[2]}-\frac {K[1] \text {F1}'\left (K[1] \left (1+\frac {1}{K[2]}\right )\right )}{K[2]^2}}{K[1] \left (K[2] \text {F1}\left (K[1] \left (1+\frac {1}{K[2]}\right )\right )+\text {F1}\left (K[1] \left (1+\frac {1}{K[2]}\right )\right )\right )-K[2]}-\frac {\left (K[2] \text {F1}\left (K[1] \left (1+\frac {1}{K[2]}\right )\right )+\text {F1}\left (K[1] \left (1+\frac {1}{K[2]}\right )\right )\right ) \left (K[1] \left (\text {F1}\left (K[1] \left (1+\frac {1}{K[2]}\right )\right )-\frac {K[1] \text {F1}'\left (K[1] \left (1+\frac {1}{K[2]}\right )\right )}{K[2]}-\frac {K[1] \text {F1}'\left (K[1] \left (1+\frac {1}{K[2]}\right )\right )}{K[2]^2}\right )-1\right )}{\left (K[1] \left (K[2] \text {F1}\left (K[1] \left (1+\frac {1}{K[2]}\right )\right )+\text {F1}\left (K[1] \left (1+\frac {1}{K[2]}\right )\right )\right )-K[2]\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\left (\frac {y(x) \text {F1}\left (K[1] \left (1+\frac {1}{y(x)}\right )\right )+\text {F1}\left (K[1] \left (1+\frac {1}{y(x)}\right )\right )}{K[1] \left (y(x) \text {F1}\left (K[1] \left (1+\frac {1}{y(x)}\right )\right )+\text {F1}\left (K[1] \left (1+\frac {1}{y(x)}\right )\right )\right )-y(x)}-\frac {1}{K[1]}\right )dK[1]=c_1,y(x)\right ] \]