54.2.237 problem 814

Internal problem ID [12111]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 814
Date solved : Wednesday, October 01, 2025 at 12:42:12 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class C`], [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }&=\frac {y \left (-3 x^{3} y-3+y^{2} x^{7}\right )}{x \left (x^{3} y+1\right )} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 42

ode:=diff(y(x),x) = y(x)/x*(-3*x^3*y(x)-3+y(x)^2*x^7)/(x^3*y(x)+1); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= \frac {1}{\left (\sqrt {c_1 -2 x}-1\right ) x^{3}} \\ y &= -\frac {1}{\left (\sqrt {c_1 -2 x}+1\right ) x^{3}} \\ \end{align*}

✓ Mathematica. Time used: 0.529 (sec). Leaf size: 75

ode=D[y[x],x] == (y[x]*(-3 - 3*x^3*y[x] + x^7*y[x]^2))/(x*(1 + x^3*y[x])); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x}{-x^4+\frac {\sqrt {x (-2 x+1+c_1)}}{\sqrt {\frac {1}{x^7}}}}\\ y(x)&\to -\frac {x}{x^4+\frac {\sqrt {x (-2 x+1+c_1)}}{\sqrt {\frac {1}{x^7}}}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 27.416 (sec). Leaf size: 73

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x**7*y(x)**2 - 3*x**3*y(x) - 3)*y(x)/(x*(x**3*y(x) + 1)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {- x^{3} - \sqrt {- x^{6} \left (2 x - e^{2 C_{1}}\right )}}{x^{6} \left (2 x - e^{2 C_{1}} + 1\right )}, \ y{\left (x \right )} = \frac {- x^{3} + \sqrt {- x^{6} \left (2 x - e^{2 C_{1}}\right )}}{x^{6} \left (2 x - e^{2 C_{1}} + 1\right )}\right ] \]