64.22.7 problem 3(c)

Internal problem ID [13601]
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(c)
Date solved : Monday, March 31, 2025 at 08:02:20 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \frac {\left (t +1\right ) x^{\prime \prime }}{t}-\frac {x^{\prime }}{t^{2}}+\frac {x}{t^{3}}&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 19

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

\[ x = t \left (c_2 \ln \left (t \right )-c_2 \ln \left (t +1\right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.163 (sec). Leaf size: 103

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

\[ x(t)\to \exp \left (\int _1^t\frac {2 K[1]+1}{2 K[1]^2+2 K[1]}dK[1]-\frac {1}{2} \int _1^t-\frac {1}{K[2]^2+K[2]}dK[2]\right ) \left (c_2 \int _1^t\exp \left (-2 \int _1^{K[3]}\frac {2 K[1]+1}{2 K[1]^2+2 K[1]}dK[1]\right )dK[3]+c_1\right ) \]

✓ Sympy. Time used: 0.767 (sec). Leaf size: 8

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq((t + 1)*Derivative(x(t), (t, 2))/t - Derivative(x(t), t)/t**2 + x(t)/t**3,0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = C_{1} t + O\left (t^{6}\right ) \]