59.1.350 problem 357

Internal problem ID [9522]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 357
Date solved : Wednesday, March 05, 2025 at 07:50:25 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\frac {\left (x^{2}-1\right ) y}{4}&=0 \end{align*}

✓ Maple. Time used: 0.037 (sec). Leaf size: 21

ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+1/4*(x^2-1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_{1} \sin \left (\frac {x}{2}\right )+c_{2} \cos \left (\frac {x}{2}\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 36

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+1/4*(x^2-1)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {e^{-\frac {i x}{2}} \left (c_1-i c_2 e^{i x}\right )}{\sqrt {x}} \]

✓ Sympy. Time used: 0.226 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + (x**2 - 1)*y(x)/4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} J_{\frac {1}{2}}\left (\frac {x}{2}\right ) + C_{2} Y_{\frac {1}{2}}\left (\frac {x}{2}\right ) \]