54.2.275 problem 853

Internal problem ID [12149]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 853
Date solved : Wednesday, October 01, 2025 at 01:00:54 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=\frac {14 x y+12+2 x +x^{3} y^{3}+6 x^{2} y^{2}}{x^{2} \left (x y+2+x \right )} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 71

ode:=diff(y(x),x) = 1/x^2*(14*x*y(x)+12+2*x+x^3*y(x)^3+6*x^2*y(x)^2)/(x*y(x)+2+x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {2}{x} \\ y &= \frac {-2 \sqrt {c_1 -2 x}+x +2}{\left (\sqrt {c_1 -2 x}-1\right ) x} \\ y &= \frac {-2 \sqrt {c_1 -2 x}-x -2}{\left (\sqrt {c_1 -2 x}+1\right ) x} \\ \end{align*}

✓ Mathematica. Time used: 0.261 (sec). Leaf size: 84

ode=D[y[x],x] == (12 + 2*x + 14*x*y[x] + 6*x^2*y[x]^2 + x^3*y[x]^3)/(x^2*(2 + x + x*y[x])); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x-2 \sqrt {-2 x+c_1}+2}{x \left (-1+\sqrt {-2 x+c_1}\right )}\\ y(x)&\to -\frac {x+2 \sqrt {-2 x+c_1}+2}{x+x \sqrt {-2 x+c_1}}\\ y(x)&\to -\frac {2}{x} \end{align*}

✓ Sympy. Time used: 1.538 (sec). Leaf size: 60

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x**3*y(x)**3 + 6*x**2*y(x)**2 + 14*x*y(x) + 2*x + 12)/(x**2*(x*y(x) + x + 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {- 4 C_{1} - x \sqrt {- 2 C_{1} - 2 x + 1} - 5 x}{2 x \left (C_{1} + x\right )}, \ y{\left (x \right )} = \frac {- 4 C_{1} + x \sqrt {- 2 C_{1} - 2 x + 1} - 5 x}{2 x \left (C_{1} + x\right )}\right ] \]