2.1.1 Problem 2

Internal problem ID [18522]
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 : Sunday, February 23, 2025 at 09:48:04 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

Solve

\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 step by step solution

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, 
   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`
 

Maple dsolve solution

Solving time : 0.010 (sec)
Leaf size : 20

dsolve(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,y(x),singsol=all)
 

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

✓Mathematica DSolve solution

Solving time : 0.245 (sec)
Leaf size : 19

DSolve[{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,{}},y[x],x,IncludeSingularSolutions->True]
 

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

✗Sympy solution

Solving time : 0.000 (sec)
Leaf size : 0

Python version: 3.13.1 (main, Dec  4 2024, 18:05:56) [GCC 14.2.1 20240910] 
Sympy version 1.13.3
 

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