82.23.2 problem Ex. 2

Internal problem ID [18787]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter IV. Singular solutions. problems on chapter IV. page 49
Problem number : Ex. 2
Date solved : Monday, March 31, 2025 at 06:14:34 PM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} x^{3} {y^{\prime }}^{2}+x^{2} y y^{\prime }+a^{3}&=0 \end{align*}

✓ Maple. Time used: 0.371 (sec). Leaf size: 72

ode:=x^3*diff(y(x),x)^2+x^2*y(x)*diff(y(x),x)+a^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {2 \sqrt {x a}\, a}{x} \\ y &= \frac {2 \sqrt {x a}\, a}{x} \\ y &= \frac {c_1^{2} x +4 a^{3}}{2 c_1 x} \\ y &= \frac {4 a^{3} x +c_1^{2}}{2 c_1 x} \\ \end{align*}

✓ Mathematica. Time used: 0.902 (sec). Leaf size: 61

ode=x^3*D[y[x],x]^2+x^2*y[x]*D[y[x],x]+a^3==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {e^{-\frac {c_1}{2}} \left (x+4 a^3 e^{c_1}\right )}{2 x} \\ y(x)\to \frac {e^{-\frac {c_1}{2}} \left (x+4 a^3 e^{c_1}\right )}{2 x} \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**2*y(x) + sqrt(x**3*(-4*a**3 + x*y(x)**2)))/(2*x**3) cannot be solved by the factorable group method