54.2.251 problem 828

Internal problem ID [12125]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 828
Date solved : Wednesday, October 01, 2025 at 12:50:36 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} y^{\prime }&=\frac {\left (1+2 y\right ) \left (y+1\right )}{x \left (-2 y-2+x y^{3}+2 x y^{4}\right )} \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 67

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

\begin{align*} y &= -1 \\ y &= -{\frac {1}{2}} \\ y &= \frac {{\mathrm e}^{\operatorname {RootOf}\left (16 x \,{\mathrm e}^{\textit {\_Z}} \ln \left (1+{\mathrm e}^{\textit {\_Z}}\right )-16 x \,{\mathrm e}^{\textit {\_Z}} \ln \left (2\right )+8 c_1 x \,{\mathrm e}^{\textit {\_Z}}-2 \textit {\_Z} x \,{\mathrm e}^{\textit {\_Z}}+7 x \,{\mathrm e}^{\textit {\_Z}}+x \,{\mathrm e}^{3 \textit {\_Z}}-8 x \,{\mathrm e}^{2 \textit {\_Z}}+16\right )}}{2}-\frac {1}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.125 (sec). Leaf size: 62

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

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {K[1]}{4}-\frac {1}{2 (K[1]+1)}+\frac {1}{8 (2 K[1]+1)}+\frac {3}{8}\right )dK[1]-\frac {1}{2 x (2 y(x)+1)}=c_1,y(x)\right ] \]

✓ Sympy. Time used: 2.168 (sec). Leaf size: 42

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)**4 + x*y(x)**3 - 2*y(x) - 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} - \frac {y^{2}{\left (x \right )}}{4} + \frac {3 y{\left (x \right )}}{4} + \frac {\log {\left (y{\left (x \right )} + \frac {1}{2} \right )}}{8} - \log {\left (y{\left (x \right )} + 1 \right )} - \frac {1}{x \left (2 y{\left (x \right )} + 1\right )} = 0 \]