63.13.5 problem 6

Internal problem ID [13097]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.4.3 Reduction of order. Exercises page 125
Problem number : 6
Date solved : Wednesday, March 05, 2025 at 09:17:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} x&=\frac {\cos \left (t \right )}{\sqrt {t}} \end{align*}

✓ Maple. Time used: 0.160 (sec). Leaf size: 17

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

\[ x \left (t \right ) = \frac {c_{2} \cos \left (t \right )+c_{1} \sin \left (t \right )}{\sqrt {t}} \]

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 39

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

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

✓ Sympy. Time used: 0.242 (sec). Leaf size: 15

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

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