54.2.226 problem 803

Internal problem ID [12100]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 803
Date solved : Sunday, October 12, 2025 at 02:16:08 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} y^{\prime }&=\frac {\textit {\_F1} \left (y^{2}-2 \ln \left (x \right )\right )}{\sqrt {y^{2}}\, x} \end{align*}

✓ Maple. Time used: 0.176 (sec). Leaf size: 65

ode:=diff(y(x),x) = _F1(y(x)^2-2*ln(x))/(y(x)^2)^(1/2)/x; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \sqrt {2 \ln \left (x \right )+2 \operatorname {RootOf}\left (\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_F1} \left (2 \textit {\_a} \right )-1}d \textit {\_a} +c_1 \right )} \\ y &= -\sqrt {2 \ln \left (x \right )+2 \operatorname {RootOf}\left (\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_F1} \left (2 \textit {\_a} \right )-1}d \textit {\_a} +c_1 \right )} \\ \end{align*}

✓ Mathematica. Time used: 0.506 (sec). Leaf size: 603

ode=D[y[x],x] == F1[-2*Log[x] + y[x]^2]/(x*Sqrt[y[x]^2]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
F1 = Function("F1") 
ode = Eq(Derivative(y(x), x) - F1(y(x)**2 - 2*log(x))/(x*sqrt(y(x)**2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - F1(y(x)**2 - 2*log(x))/(x*sqrt(y(x)**2)) cannot be solved by the lie group method