60.2.155 problem 731

Internal problem ID [10729]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 731
Date solved : Wednesday, March 05, 2025 at 12:29:07 PM
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}+2 x y^{3}\right )} \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 46

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

\begin{align*} y &= -{\frac {1}{2}} \\ y &= \frac {{\mathrm e}^{\operatorname {RootOf}\left ({\mathrm e}^{3 \textit {\_Z}} x -4 x \,{\mathrm e}^{2 \textit {\_Z}}+8 c_{1} x \,{\mathrm e}^{\textit {\_Z}}+2 \textit {\_Z} x \,{\mathrm e}^{\textit {\_Z}}+3 x \,{\mathrm e}^{\textit {\_Z}}+16\right )}}{2}-\frac {1}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.171 (sec). Leaf size: 52

ode=D[y[x],x] == (1 + 2*y[x])/(x*(-2 + x*y[x]^2 + 2*x*y[x]^3)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {K[1]}{8}-\frac {1}{16 (2 K[1]+1)}+\frac {1}{16}\right )dK[1]-\frac {1}{4 x (2 y(x)+1)}=c_1,y(x)\right ] \]

✓ Sympy. Time used: 1.443 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (2*y(x) + 1)/(x*(2*x*y(x)**3 + x*y(x)**2 - 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} - \frac {y^{2}{\left (x \right )}}{4} + \frac {y{\left (x \right )}}{4} - \frac {\log {\left (2 y{\left (x \right )} + 1 \right )}}{8} - \frac {1}{x \left (2 y{\left (x \right )} + 1\right )} = 0 \]