59.1.279 problem 282

Internal problem ID [9451]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 282
Date solved : Sunday, March 30, 2025 at 02:35:14 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 17

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

\[ y = \frac {c_1 \sinh \left (x \right )+c_2 \cosh \left (x \right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 32

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

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

✓ Sympy. Time used: 0.230 (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 + 1/4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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