58.2.36 problem 36

Internal problem ID [9159]
Book : Second order enumerated odes
Section : section 2
Problem number : 36
Date solved : Thursday, March 13, 2025 at 06:58:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.036 (sec). Leaf size: 22

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

\[ y = \left (-{\mathrm e}^{c_{1}} \operatorname {Ei}_{1}\left (-\ln \left (\frac {1}{x}\right )+c_{1} \right )+c_{2} \right ) x \]

✓ Mathematica. Time used: 28.572 (sec). Leaf size: 33

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

\begin{align*} y(x)\to x \left (e^{c_1} \operatorname {ExpIntegralEi}(-c_1-\log (x))+c_2\right ) \\ y(x)\to c_2 x \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*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(-Derivative(y(x), (x, 2))) + Derivative(y(x), x) - y(x)/x cannot be solved by the factorable group method