60.2.309 problem 887

Internal problem ID [10883]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 887
Date solved : Wednesday, March 05, 2025 at 01:15:11 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class C`]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 72

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

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

✓ Mathematica. Time used: 0.91 (sec). Leaf size: 103

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

\begin{align*} y(x)\to -\frac {1}{a x}+\frac {a^3}{-a^3+\sqrt {-2 a^6 x+c_1}} \\ y(x)\to -\frac {\sqrt {-2 a^6 x+c_1}+a^4 x+a^3}{a^4 x+a x \sqrt {-2 a^6 x+c_1}} \\ y(x)\to -\frac {1}{a x} \\ \end{align*}

✓ Sympy. Time used: 3.187 (sec). Leaf size: 70

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

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