23.3.120 problem 122

Internal problem ID [5834]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 122
Date solved : Tuesday, September 30, 2025 at 02:03:52 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 33

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

\[ y = \left (c_1 \,{\mathrm e}^{2 x \sqrt {-a}}+c_2 \right ) {\mathrm e}^{\frac {x \left (a x -2 \sqrt {-a}\right )}{2}} \]

✓ Mathematica. Time used: 0.038 (sec). Leaf size: 63

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

\begin{align*} y(x)&\to \frac {e^{\frac {a x^2}{2}-i \sqrt {a} x} \left (2 \sqrt {a} c_1-i c_2 e^{2 i \sqrt {a} x}\right )}{2 \sqrt {a}} \end{align*}

✗ Sympy

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

False