54.2.294 problem 873

Internal problem ID [12168]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 873
Date solved : Wednesday, October 01, 2025 at 01:05:35 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 +x y^{2}+3 x y^{3}+2 x y+2 x y^{4}\right )} \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 54

ode:=diff(y(x),x) = 1/x*(1+2*y(x))/(-2+x+x*y(x)^2+3*x*y(x)^3+2*x*y(x)+2*x*y(x)^4); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -{\frac {1}{2}} \\ y &= \frac {{\mathrm e}^{\operatorname {RootOf}\left (2 x \,{\mathrm e}^{4 \textit {\_Z}}-3 x \,{\mathrm e}^{3 \textit {\_Z}}-6 x \,{\mathrm e}^{2 \textit {\_Z}}+48 c_1 x \,{\mathrm e}^{\textit {\_Z}}+54 \textit {\_Z} x \,{\mathrm e}^{\textit {\_Z}}+7 x \,{\mathrm e}^{\textit {\_Z}}+96\right )}}{2}-\frac {1}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.09 (sec). Leaf size: 60

ode=D[y[x],x] == (1 + 2*y[x])/(x*(-2 + x + 2*x*y[x] + x*y[x]^2 + 3*x*y[x]^3 + 2*x*y[x]^4)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {1}{4} K[1]^2-\frac {K[1]}{8}-\frac {9}{16 (2 K[1]+1)}+\frac {1}{16}\right )dK[1]-\frac {1}{2 x (2 y(x)+1)}=c_1,y(x)\right ] \]

✓ Sympy. Time used: 1.353 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (2*y(x) + 1)/(x*(2*x*y(x)**4 + 3*x*y(x)**3 + x*y(x)**2 + 2*x*y(x) + x - 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} - \frac {y^{3}{\left (x \right )}}{6} - \frac {y^{2}{\left (x \right )}}{8} + \frac {y{\left (x \right )}}{8} - \frac {9 \log {\left (2 y{\left (x \right )} + 1 \right )}}{16} - \frac {1}{x \left (2 y{\left (x \right )} + 1\right )} = 0 \]