15.19.23 problem 23

Internal problem ID [3307]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 37, page 171
Problem number : 23
Date solved : Sunday, March 30, 2025 at 01:34:23 AM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Maple. Time used: 0.300 (sec). Leaf size: 51

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

\begin{align*} y &= -\frac {2 x^{{3}/{2}}}{3} \\ y &= \frac {2 x^{{3}/{2}}}{3} \\ y &= \frac {c_1^{2} x^{3}+4}{6 c_1} \\ y &= \frac {4 x^{3}+c_1^{2}}{6 c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.341 (sec). Leaf size: 83

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

\begin{align*} y(x)\to -\frac {1}{6} e^{-\frac {3 c_1}{2}} \left (4 x^3+e^{3 c_1}\right ) \\ y(x)\to \frac {1}{6} e^{-\frac {3 c_1}{2}} \left (4 x^3+e^{3 c_1}\right ) \\ y(x)\to -\frac {2 x^{3/2}}{3} \\ y(x)\to \frac {2 x^{3/2}}{3} \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt(-4*x**3 + 9*y(x)**2) + 3*y(x))/(2*x) cannot be solved by the factorable group method