54.2.406 problem 985

Internal problem ID [12280]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 985
Date solved : Wednesday, October 01, 2025 at 01:27:46 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Abel]

\begin{align*} y^{\prime }&=\frac {\left (x y+1\right ) \left (x^{2} y^{2}+x^{2} y+2 x y+1+x +x^{2}\right )}{x^{5}} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 43

ode:=diff(y(x),x) = (x*y(x)+1)*(x^2*y(x)^2+x^2*y(x)+2*x*y(x)+1+x+x^2)/x^5; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {17 \operatorname {RootOf}\left (162 \int _{}^{\textit {\_Z}}\frac {1}{289 \textit {\_a}^{3}+54 \textit {\_a} -54}d \textit {\_a} x +3 c_1 x +2\right ) x -3 x -9}{9 x} \]

✓ Mathematica. Time used: 0.163 (sec). Leaf size: 86

ode=D[y[x],x] == ((1 + x*y[x])*(1 + x + x^2 + 2*x*y[x] + x^2*y[x] + x^2*y[x]^2))/x^5; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{\frac {\frac {x+3}{x^3}+\frac {3 y(x)}{x^2}}{\sqrt [3]{34} \sqrt [3]{-\frac {1}{x^6}}}}\frac {1}{K[1]^3-\frac {3 \sqrt [3]{-2} K[1]}{17^{2/3}}+1}dK[1]=-\frac {1}{9} 34^{2/3} \left (-\frac {1}{x^6}\right )^{2/3} x^3+c_1,y(x)\right ] \]

✓ Sympy. Time used: 144.792 (sec). Leaf size: 1722

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x*y(x) + 1)*(x**2*y(x)**2 + x**2*y(x) + x**2 + 2*x*y(x) + x + 1)/x**5,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \text {Solution too large to show} \]