59.1.677 problem 694

Internal problem ID [9849]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 694
Date solved : Wednesday, March 05, 2025 at 07:59:55 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {9}{4}\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.050 (sec). Leaf size: 30

ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+(x^2-9/4)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-\left (-x +i\right ) c_{2} {\mathrm e}^{-i x}+\left (x +i\right ) c_{1} {\mathrm e}^{i x}}{x^{{3}/{2}}} \]

✓ Mathematica. Time used: 0.059 (sec). Leaf size: 44

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+(x^2-9/4)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -\frac {\sqrt {\frac {2}{\pi }} ((c_1 x+c_2) \cos (x)+(c_2 x-c_1) \sin (x))}{x^{3/2}} \]

✓ Sympy. Time used: 0.221 (sec). Leaf size: 15

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 - 9/4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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