59.1.349 problem 356

Internal problem ID [9521]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 356
Date solved : Wednesday, March 05, 2025 at 07:50:24 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\frac {\left (x +\frac {3}{4}\right ) y}{4}&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 21

ode:=x^2*diff(diff(y(x),x),x)+1/4*(x+3/4)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x^{{1}/{4}} \left (c_{1} \sin \left (\sqrt {x}\right )+c_{2} \cos \left (\sqrt {x}\right )\right ) \]

✓ Mathematica. Time used: 0.06 (sec). Leaf size: 43

ode=x^2*D[y[x],{x,2}]+1/4*(x+3/4)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{-i \sqrt {x}} \sqrt [4]{x} \left (c_1 e^{2 i \sqrt {x}}+i c_2\right ) \]

✓ Sympy. Time used: 0.090 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + (x + 3/4)*y(x)/4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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