64.22.5 problem 3(a)

Internal problem ID [13599]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 11, The nth order homogeneous linear differential equation. Section 11.8, Exercises page 583
Problem number : 3(a)
Date solved : Monday, March 31, 2025 at 08:02:16 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} t^{3} x^{\prime \prime }+3 t^{2} x^{\prime }+x&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 53

ode:=t^3*diff(diff(x(t),t),t)+3*t^2*diff(x(t),t)+x(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

\[ x = \frac {\operatorname {BesselY}\left (1, \frac {2}{\sqrt {t}}\right ) \sqrt {t}\, c_2 +\operatorname {BesselJ}\left (1, \frac {2}{\sqrt {t}}\right ) \sqrt {t}\, c_1 -\operatorname {BesselY}\left (0, \frac {2}{\sqrt {t}}\right ) c_2 -\operatorname {BesselJ}\left (0, \frac {2}{\sqrt {t}}\right ) c_1}{t} \]

✓ Mathematica. Time used: 0.055 (sec). Leaf size: 40

ode=t^3*D[x[t],{t,2}]+3*t^2*D[x[t],t]+x[t]==0; 
ic={}; 
DSolve[{ode,ic},{x[t]},t,IncludeSingularSolutions->True]
 

\[ x(t)\to \frac {2 c_1 \operatorname {BesselJ}\left (2,2 \sqrt {\frac {1}{t}}\right )-2 c_2 \operatorname {BesselY}\left (2,2 \sqrt {\frac {1}{t}}\right )}{t} \]

✓ Sympy. Time used: 0.202 (sec). Leaf size: 26

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

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