54.2.402 problem 981

Internal problem ID [12276]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 981
Date solved : Wednesday, October 01, 2025 at 01:26:51 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Abel]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 52

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

\begin{align*} y &= -\frac {1}{x a} \\ y &= -\frac {1}{\sqrt {c_1 -2 x}}-\frac {1}{x a} \\ y &= \frac {1}{\sqrt {c_1 -2 x}}-\frac {1}{x a} \\ \end{align*}

✓ Mathematica. Time used: 0.181 (sec). Leaf size: 61

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

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

✓ Sympy. Time used: 1.287 (sec). Leaf size: 63

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 + 3*a*x*y(x) + 1)/(a**3*x**3),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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