54.2.162 problem 739

Internal problem ID [12036]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 739
Date solved : Wednesday, October 01, 2025 at 12:01:27 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 39

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

\begin{align*} y &= -{\frac {1}{2}} \\ y &= \frac {{\mathrm e}^{\operatorname {RootOf}\left (x \,{\mathrm e}^{2 \textit {\_Z}}+2 c_1 x \,{\mathrm e}^{\textit {\_Z}}-\textit {\_Z} x \,{\mathrm e}^{\textit {\_Z}}-x \,{\mathrm e}^{\textit {\_Z}}+4\right )}}{2}-\frac {1}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.103 (sec). Leaf size: 46

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

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {1}{4 (2 K[1]+1)}-\frac {1}{4}\right )dK[1]-\frac {1}{2 x (2 y(x)+1)}=c_1,y(x)\right ] \]

✓ Sympy. Time used: 1.002 (sec). Leaf size: 27

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

\[ C_{1} - \frac {y{\left (x \right )}}{2} + \frac {\log {\left (2 y{\left (x \right )} + 1 \right )}}{4} - \frac {1}{x \left (2 y{\left (x \right )} + 1\right )} = 0 \]