60.2.226 problem 802

Internal problem ID [10813]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 802
Date solved : Tuesday, January 28, 2025 at 05:16:40 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

\begin{align*} y^{\prime }&=-\frac {-\frac {1}{x}-\textit {\_F1} \left (y+\frac {1}{x}\right )}{x} \end{align*}

✓ Solution by Maple

Time used: 0.038 (sec). Leaf size: 41

dsolve(diff(y(x),x) = -(-1/x-_F1(y(x)+1/x))/x,y(x), singsol=all)
 

\begin{align*} y &= \frac {\operatorname {RootOf}\left (f_{1} \left (\textit {\_Z} \right )\right ) x -1}{x} \\ y &= \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {1}{f_{1} \left (\textit {\_a} \right )}d \textit {\_a} +c_{1} \right ) x -1}{x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.178 (sec). Leaf size: 96

DSolve[D[y[x],x] == (x^(-1) + F1[x^(-1) + y[x]])/x,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}-\frac {\text {F1}\left (K[2]+\frac {1}{x}\right ) \int _1^x-\frac {\text {F1}''\left (K[2]+\frac {1}{K[1]}\right )}{\text {F1}\left (K[2]+\frac {1}{K[1]}\right )^2 K[1]^2}dK[1]+1}{\text {F1}\left (K[2]+\frac {1}{x}\right )}dK[2]+\int _1^x\left (\frac {1}{K[1]}+\frac {1}{K[1]^2 \text {F1}\left (y(x)+\frac {1}{K[1]}\right )}\right )dK[1]=c_1,y(x)\right ] \]