60.2.51 problem 627

Internal problem ID [10625]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 627
Date solved : Wednesday, March 05, 2025 at 12:11:35 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 35

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

\[ y = \frac {\sin \left (\ln \left (x \right )\right ) c_{1} +\cos \left (\ln \left (x \right )\right )}{\left (\ln \left (x \right )+c_{1} \right ) \cos \left (\ln \left (x \right )\right )+\left (\ln \left (x \right ) c_{1} -1\right ) \sin \left (\ln \left (x \right )\right )} \]

✓ Mathematica. Time used: 0.901 (sec). Leaf size: 63

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

\begin{align*} y(x)\to \frac {\sin (\log (x))+c_1 \cos (\log (x))}{(1+c_1 \log (x)) \cos (\log (x))+(\log (x)-c_1) \sin (\log (x))} \\ y(x)\to \frac {\cos (\log (x))}{\log (x) \cos (\log (x))-\sin (\log (x))} \\ \end{align*}

✗ Sympy

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

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