54.2.124 problem 700

Internal problem ID [11998]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 700
Date solved : Tuesday, September 30, 2025 at 11:52:40 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 62

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

\begin{align*} y &= \frac {\sqrt {x \left (2 \operatorname {LambertW}\left (\frac {c_1 \,{\mathrm e}^{-\frac {x -1}{2 x}}}{2}\right ) x +x -1\right )}}{x} \\ y &= -\frac {\sqrt {x \left (2 \operatorname {LambertW}\left (\frac {c_1 \,{\mathrm e}^{-\frac {x -1}{2 x}}}{2}\right ) x +x -1\right )}}{x} \\ \end{align*}

✓ Mathematica. Time used: 60.285 (sec). Leaf size: 72

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

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

✓ Sympy. Time used: 0.783 (sec). Leaf size: 31

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

\[ C_{1} - \left (y^{2}{\left (x \right )} - 1\right ) \sqrt {e^{y^{2}{\left (x \right )}}} - \frac {\sqrt {e^{y^{2}{\left (x \right )}}}}{x} = 0 \]