54.3.48 problem 1053

Internal problem ID [12343]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1053
Date solved : Wednesday, October 01, 2025 at 01:43:30 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 29

ode:=diff(diff(y(x),x),x)+2*a*x*diff(y(x),x)+a^2*x^2*y(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: 0.041 (sec). Leaf size: 56

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}-\sqrt {a} x} \left (c_2 e^{2 \sqrt {a} x}+2 \sqrt {a} c_1\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