Internal problem ID [8441]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 861.
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 {\left (-\frac {y \,{\mathrm e}^{\frac {1}{x}}}{x}-\textit {\_F1} \left (y \,{\mathrm e}^{\frac {1}{x}}\right )\right ) {\mathrm e}^{-\frac {1}{x}}}{x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 39
dsolve(diff(y(x),x) = -(-1/x*y(x)/exp(-1/x)-_F1(y(x)/exp(-1/x)))*exp(-1/x)/x,y(x), singsol=all)
\begin{align*} y \relax (x ) = \RootOf \left (\textit {\_F1} \left (\textit {\_Z} \right )\right ) {\mathrm e}^{-\frac {1}{x}} \\ y \relax (x ) = \RootOf \left (-\ln \relax (x )+\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_F1} \left (\textit {\_a} \right )}d \textit {\_a} +c_{1}\right ) {\mathrm e}^{-\frac {1}{x}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.818 (sec). Leaf size: 152
DSolve[y'[x] == (F1[E^x^(-1)*y[x]] + (E^x^(-1)*y[x])/x)/(E^x^(-1)*x),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}-\frac {\text {F1}\left (e^{\frac {1}{x}} K[2]\right ) \int _1^x\left (\frac {e^{\frac {1}{K[1]}}}{\text {F1}\left (e^{\frac {1}{K[1]}} K[2]\right ) K[1]^2}-\frac {e^{\frac {2}{K[1]}} K[2] \text {F1}'\left (e^{\frac {1}{K[1]}} K[2]\right )}{\text {F1}\left (e^{\frac {1}{K[1]}} K[2]\right )^2 K[1]^2}\right )dK[1]+e^{\frac {1}{x}}}{\text {F1}\left (e^{\frac {1}{x}} K[2]\right )}dK[2]+\int _1^x\left (\frac {e^{\frac {1}{K[1]}} y(x)}{\text {F1}\left (e^{\frac {1}{K[1]}} y(x)\right ) K[1]^2}+\frac {1}{K[1]}\right )dK[1]=c_1,y(x)\right ] \]