54.2.231 problem 808

Internal problem ID [12105]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 808
Date solved : Wednesday, October 01, 2025 at 12:35:42 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.044 (sec). Leaf size: 45

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

\[ y = \frac {-x \operatorname {LambertW}\left (\frac {{\mathrm e}^{-\frac {1}{x}}}{x c_1}\right )-2}{2 x \operatorname {LambertW}\left (\frac {{\mathrm e}^{-\frac {1}{x}}}{x c_1}\right )+2} \]

✓ Mathematica. Time used: 0.171 (sec). Leaf size: 75

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

\[ \text {Solve}\left [\int _1^{-\frac {2^{2/3} (2 y(x) x+x+y(x)+1)}{x+2 (x-1) y(x)-2}}\frac {1}{K[1]^3-\frac {3 K[1]}{2^{2/3}}+1}dK[1]+\frac {1}{9} 2^{2/3} \left (\frac {1}{x}+\log (x)-1\right )=c_1,y(x)\right ] \]

✓ Sympy. Time used: 1.076 (sec). Leaf size: 27

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

\[ C_{1} - \log {\left (y{\left (x \right )} + \frac {1}{2} \right )} + \log {\left (y{\left (x \right )} + 1 \right )} - \frac {1}{x \left (2 y{\left (x \right )} + 1\right )} = 0 \]