53.1.384 problem 394

Internal problem ID [10856]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 394
Date solved : Tuesday, September 30, 2025 at 07:32:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x y^{\prime }-\left (x^{2}+\frac {5}{4}\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 25

ode:=x^2*diff(diff(y(x),x),x)-x*diff(y(x),x)-(x^2+5/4)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-x} c_2 \left (x +1\right )+{\mathrm e}^{x} c_1 \left (x -1\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 53

ode=x^2*D[y[x],{x,2}]-x*D[y[x],x]-(x^2+5/4)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\sqrt {\frac {2}{\pi }} ((i c_2 x+c_1) \sinh (x)-(c_1 x+i c_2) \cosh (x))}{\sqrt {-i x}} \end{align*}

✓ Sympy. Time used: 0.132 (sec). Leaf size: 20

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 + 5/4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = x \left (C_{1} J_{\frac {3}{2}}\left (i x\right ) + C_{2} Y_{\frac {3}{2}}\left (i x\right )\right ) \]