61.29.20 problem 129

Internal problem ID [12550]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-4
Problem number : 129
Date solved : Monday, March 31, 2025 at 05:38:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-2 a x y^{\prime }+\left (b^{2} x^{2}+a \left (a +1\right )\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 21

ode:=x^2*diff(diff(y(x),x),x)-2*a*x*diff(y(x),x)+(b^2*x^2+a*(a+1))*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x^{a} \left (c_1 \sin \left (b x \right )+c_2 \cos \left (b x \right )\right ) \]

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 42

ode=x^2*D[y[x],{x,2}]-2*a*x*D[y[x],x]+(b^2*x^2+a*(a+1))*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 x^a e^{-i b x}-\frac {i c_2 x^a e^{i b x}}{2 b} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(-2*a*x*Derivative(y(x), x) + x**2*Derivative(y(x), (x, 2)) + (a*(a + 1) + b**2*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : invalid input: 2*a + 1