54.2.283 problem 861

Internal problem ID [12157]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 861
Date solved : Sunday, October 12, 2025 at 02:24:52 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} 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} \end{align*}

✓ Maple. Time used: 0.092 (sec). Leaf size: 39

ode:=diff(y(x),x) = -(-1/x*y(x)/exp(-1/x)-_F1(y(x)/exp(-1/x)))*exp(-1/x)/x; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \operatorname {RootOf}\left (\textit {\_F1} \left (\textit {\_Z} \right )\right ) {\mathrm e}^{-\frac {1}{x}} \\ y &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_F1} \left (\textit {\_a} \right )}d \textit {\_a} +c_1 \right ) {\mathrm e}^{-\frac {1}{x}} \\ \end{align*}

✓ Mathematica. Time used: 0.41 (sec). Leaf size: 152

ode=D[y[x],x] == (F1[E^x^(-1)*y[x]] + (E^x^(-1)*y[x])/x)/(E^x^(-1)*x); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
F1 = Function("F1") 
ode = Eq(Derivative(y(x), x) + (-F1(y(x)*exp(1/x)) - y(x)*exp(1/x)/x)*exp(-1/x)/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - F1(y(x)*exp(1/x))*exp(-1/x)/x - y(x)/x**2 cannot be solved by the factorable group method