54.2.414 problem 993

Internal problem ID [12288]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 993
Date solved : Wednesday, October 01, 2025 at 01:28:10 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=-F \left (x \right ) \left (-y^{2}-2 y \ln \left (x \right )-\ln \left (x \right )^{2}\right )+\frac {y}{\ln \left (x \right ) x} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 33

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

\[ y = -\frac {\ln \left (x \right ) \left (2 \int \ln \left (x \right ) F \left (x \right )d x +c_1 +2\right )}{2 \int \ln \left (x \right ) F \left (x \right )d x +c_1} \]

✓ Mathematica. Time used: 1.389 (sec). Leaf size: 75

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

\begin{align*} y(x)&\to \frac {\int _1^x\frac {F(K[1])}{\sqrt {\frac {1}{\log ^2(K[1])}}}dK[1]-1+c_1}{\sqrt {\frac {1}{\log ^2(x)}} \left (\int _1^x\frac {F(K[1])}{\sqrt {\frac {1}{\log ^2(K[1])}}}dK[1]+c_1\right )}\\ y(x)&\to \frac {1}{\sqrt {\frac {1}{\log ^2(x)}}} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (x*(y(x)**2 + 2*y(x)*log(x) + log(x)**2)*F(x)*log(x) + y(x))/(x*log(x)) cannot be solved by the factorable group method