80.1.1 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]]

\begin{align*} x^{2} y^{\prime \prime }-\frac {x^{2} {y^{\prime }}^{2}}{2 y}+4 x y^{\prime }+4 y&=0 \end{align*}

✓ 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*x*diff(y(x),x)+4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (c_{1} x +\frac {c_{2}}{2}\right )^{2}}{c_{1} x^{4}} \]

✓ 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]
 

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

✗ 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