54.2.53 problem 629

Internal problem ID [11927]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 629
Date solved : Tuesday, September 30, 2025 at 11:43:53 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=\frac {\left (-1+2 y \ln \left (x \right )\right )^{2}}{x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 62

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

\[ y = \frac {\sin \left (\ln \left (x \right ) \sqrt {2}\right ) c_1 +\cos \left (\ln \left (x \right ) \sqrt {2}\right )}{\sin \left (\ln \left (x \right ) \sqrt {2}\right ) \left (2 \ln \left (x \right ) c_1 -\sqrt {2}\right )+\left (\sqrt {2}\, c_1 +2 \ln \left (x \right )\right ) \cos \left (\ln \left (x \right ) \sqrt {2}\right )} \]

✓ Mathematica. Time used: 0.574 (sec). Leaf size: 123

ode=D[y[x],x] == (-1 + 2*Log[x]*y[x])^2/x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\sin \left (\sqrt {2} \log (x)\right )+c_1 \cos \left (\sqrt {2} \log (x)\right )}{\left (\sqrt {2}+2 c_1 \log (x)\right ) \cos \left (\sqrt {2} \log (x)\right )+\left (2 \log (x)-\sqrt {2} c_1\right ) \sin \left (\sqrt {2} \log (x)\right )}\\ y(x)&\to \frac {\cos \left (\sqrt {2} \log (x)\right )}{2 \log (x) \cos \left (\sqrt {2} \log (x)\right )-\sqrt {2} \sin \left (\sqrt {2} \log (x)\right )} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (2*y(x)*log(x) - 1)**2/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (4*y(x)**2*log(x)**2 - 4*y(x)*log(x) + 1)/x cannot be solved by the factorable group method