2.1.1 Problem 2

Internal problem ID [18460]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter 1. section 5. Problems at page 19
Problem number : 2
Date solved : Monday, March 31, 2025 at 05:29:55 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

✓ Maple. Time used: 0.021 (sec). Leaf size: 20

ode:=x^2*diff(diff(y(x),x),x)-1/2*x^2/y(x)*diff(y(x),x)^2+4*diff(y(x),x)*x+4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

$$y=\frac{{(c_{1}x+\frac{c_2}{2})}^2}{c_1{x}^4}$$

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying 2nd order Liouville 
trying 2nd order WeierstrassP 
trying 2nd order JacobiSN 
differential order: 2; trying a linearization to 3rd order 
trying 2nd order ODE linearizable_by_differentiation 
-> Calling odsolve with the ODE, diff(diff(diff(y(x),x),x),x)+12*(x^2*diff(diff 
(y(x),x),x)+3*x*diff(y(x),x)+2*y(x))/x^3, y(x) 
   *** Sublevel 2 *** 
   Methods for third order ODEs: 
   --- Trying classification methods --- 
   trying a quadrature 
   checking if the LODE has constant coefficients 
   checking if the LODE is of Euler type 
   <- LODE of Euler type successful 
<- 2nd order ODE linearizable_by_differentiation successful
 

✓ Mathematica. Time used: 0.245 (sec). Leaf size: 19

ode=x^2*D[y[x],{x,2}]-x^2/(2*y[x])*D[y[x],x]^2+4*x*D[y[x],x]+4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

$$y(x)\to \frac{c_2(x+2c_1){}^2}{x^4}$$

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - x**2*Derivative(y(x), x)**2/(2*y(x)) + 4*x*Derivative(y(x), x) + 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt(2)*sqrt((x**2*Derivative(y(x), (x, 2)) + 12*y(x))*y(x)) + 4*y(x))/x cannot be solved by the factorable group method