13.9.13 problem 16

Internal problem ID [2399]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.2.2, Equal roots, reduction of order. Page 147
Problem number : 16
Date solved : Tuesday, March 04, 2025 at 02:09:35 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 17

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

\[ y = \frac {c_1 \sin \left (t \right )+c_2 \cos \left (t \right )}{\sqrt {t}} \]

✓ Mathematica. Time used: 0.047 (sec). Leaf size: 39

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

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

✓ Sympy. Time used: 0.225 (sec). Leaf size: 15

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

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