62.37.1 problem Ex 1

Internal problem ID [12937]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 61. Transformation of variables. Page 143
Problem number : Ex 1
Date solved : Monday, March 31, 2025 at 07:27:30 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x^{2} y y^{\prime \prime }+\left (x y^{\prime }-y\right )^{2}&=0 \end{align*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 44

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

\begin{align*} y &= 0 \\ y &= \sqrt {2}\, \sqrt {-x \left (c_1 x -c_2 \right )} \\ y &= -\sqrt {2}\, \sqrt {-x \left (c_1 x -c_2 \right )} \\ \end{align*}

✓ Mathematica. Time used: 0.257 (sec). Leaf size: 23

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

\[ y(x)\to c_2 \sqrt {x} \sqrt {2 x+c_1} \]

✗ Sympy

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

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