60.2.201 problem 777

Internal problem ID [10775]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 777
Date solved : Sunday, March 30, 2025 at 06:36:36 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 59

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

\begin{align*} y &= 0 \\ y &= -1 \\ y &= {\mathrm e}^{\operatorname {RootOf}\left ({\mathrm e}^{3 \textit {\_Z}} x -5 \,{\mathrm e}^{2 \textit {\_Z}} x +2 c_1 x \,{\mathrm e}^{\textit {\_Z}}+2 \textit {\_Z} x \,{\mathrm e}^{\textit {\_Z}}+7 \,{\mathrm e}^{\textit {\_Z}} x -2 c_1 x -2 \textit {\_Z} x -3 x +2\right )}-1 \\ \end{align*}

✓ Mathematica. Time used: 0.139 (sec). Leaf size: 38

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

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

✓ Sympy. Time used: 1.119 (sec). Leaf size: 26

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

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