68.12.56 problem 63

Internal problem ID [17650]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.4, page 163
Problem number : 63
Date solved : Thursday, October 02, 2025 at 02:26:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 t^{2} y^{\prime \prime }+4 t y^{\prime }+\left (16 t^{2}-1\right ) y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\frac {\sin \left (2 t \right )}{\sqrt {t}} \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 21

ode:=4*t^2*diff(diff(y(t),t),t)+4*t*diff(y(t),t)+(16*t^2-1)*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {c_1 \sin \left (2 t \right )+c_2 \cos \left (2 t \right )}{\sqrt {t}} \]

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 39

ode=4*t^2*D[y[t],{t,2}]+4*t*D[y[t],t]+(16*t^2-1)*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {e^{-2 i t} \left (4 c_1-i c_2 e^{4 i t}\right )}{4 \sqrt {t}} \end{align*}

✓ Sympy. Time used: 0.140 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*t**2*Derivative(y(t), (t, 2)) + 4*t*Derivative(y(t), t) + (16*t**2 - 1)*y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

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