Internal problem ID [8382]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 802.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)]]]
Solve \begin {gather*} \boxed {y^{\prime }+\frac {-\frac {1}{x}-\textit {\_F1} \left (y+\frac {1}{x}\right )}{x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.046 (sec). Leaf size: 41
dsolve(diff(y(x),x) = -(-1/x-_F1(y(x)+1/x))/x,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\RootOf \left (\textit {\_F1} \left (\textit {\_Z} \right )\right ) x -1}{x} \\ y \relax (x ) = \frac {\RootOf \left (-\ln \relax (x )+\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_F1} \left (\textit {\_a} \right )}d \textit {\_a} +c_{1}\right ) x -1}{x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.168 (sec). Leaf size: 96
DSolve[y'[x] == (x^(-1) + F1[x^(-1) + y[x]])/x,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}-\frac {\text {F1}\left (K[2]+\frac {1}{x}\right ) \int _1^x-\frac {\text {F1}'\left (K[2]+\frac {1}{K[1]}\right )}{\text {F1}\left (K[2]+\frac {1}{K[1]}\right )^2 K[1]^2}dK[1]+1}{\text {F1}\left (K[2]+\frac {1}{x}\right )}dK[2]+\int _1^x\left (\frac {1}{K[1]}+\frac {1}{K[1]^2 \text {F1}\left (y(x)+\frac {1}{K[1]}\right )}\right )dK[1]=c_1,y(x)\right ] \]