60.1.443 problem 455

Internal problem ID [10457]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 455
Date solved : Wednesday, March 05, 2025 at 10:53:58 AM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Maple. Time used: 0.190 (sec). Leaf size: 66

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

\begin{align*} y &= -\frac {2 \sqrt {a x}}{x} \\ y &= \frac {2 \sqrt {a x}}{x} \\ y &= \frac {c_{1}^{2} x +4 a}{2 x c_{1}} \\ y &= \frac {4 a x +c_{1}^{2}}{2 x c_{1}} \\ \end{align*}

✓ Mathematica. Time used: 0.605 (sec). Leaf size: 57

ode=a + x^2*y[x]*D[y[x],x] + x^3*D[y[x],x]^2==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 e^{c_1}\right )}{2 x} \\ y(x)\to \frac {e^{-\frac {c_1}{2}} \left (x+4 a e^{c_1}\right )}{2 x} \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a + 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 + x*y(x)**2)))/(2*x**3) cannot be solved by the factorable group method