54.2.361 problem 940

Internal problem ID [12235]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 940
Date solved : Wednesday, October 01, 2025 at 01:16:14 AM
CAS classification : [[_Abel, `2nd type`, `class C`], [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }&=\frac {y \ln \left (x \right ) x +x^{2} \ln \left (x \right )-2 x y-x^{2}-y^{2}-y^{3}+3 x y^{2} \ln \left (x \right )-3 x^{2} \ln \left (x \right )^{2} y+x^{3} \ln \left (x \right )^{3}}{x \left (-y+x \ln \left (x \right )-x \right )} \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 70

ode:=diff(y(x),x) = 1/x*(y(x)*ln(x)*x+x^2*ln(x)-2*x*y(x)-x^2-y(x)^2-y(x)^3+3*x*y(x)^2*ln(x)-3*x^2*ln(x)^2*y(x)+x^3*ln(x)^3)/(-y(x)+x*ln(x)-x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= x \ln \left (x \right ) \\ y &= \frac {x \left (\sqrt {c_1 -2 x}\, \ln \left (x \right )-\ln \left (x \right )+1\right )}{\sqrt {c_1 -2 x}-1} \\ y &= \frac {x \left (\sqrt {c_1 -2 x}\, \ln \left (x \right )+\ln \left (x \right )-1\right )}{\sqrt {c_1 -2 x}+1} \\ \end{align*}

✓ Mathematica. Time used: 0.349 (sec). Leaf size: 57

ode=D[y[x],x] == (-x^2 + x^2*Log[x] + x^3*Log[x]^3 - 2*x*y[x] + x*Log[x]*y[x] - 3*x^2*Log[x]^2*y[x] - y[x]^2 + 3*x*Log[x]*y[x]^2 - y[x]^3)/(x*(-x + x*Log[x] - y[x])); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x \left (\log (x)-\frac {1}{1+\sqrt {-2 x+c_1}}\right )\\ y(x)&\to x \left (\log (x)+\frac {1}{-1+\sqrt {-2 x+c_1}}\right )\\ y(x)&\to x \log (x) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x**3*log(x)**3 - 3*x**2*y(x)*log(x)**2 + x**2*log(x) - x**2 + 3*x*y(x)**2*log(x) + x*y(x)*log(x) - 2*x*y(x) - y(x)**3 - y(x)**2)/(x*(x*log(x) - x - y(x))),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out