54.2.233 problem 810

Internal problem ID [12107]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 810
Date solved : Wednesday, October 01, 2025 at 12:40:03 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 16

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

\[ y = \left (\ln \left (x \right )+\frac {1}{c_1 -x}\right ) x \]

✓ Mathematica. Time used: 0.13 (sec). Leaf size: 26

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

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

✓ Sympy. Time used: 0.963 (sec). Leaf size: 24

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

\[ y{\left (x \right )} = \frac {C_{1} x \log {\left (x^{x} \right )} - C_{1} x - \log {\left (x^{x} \right )}}{C_{1} x - 1} \]