64.21.4 problem 6

Internal problem ID [13586]
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 11.6, Exercises page 567
Problem number : 6
Date solved : Friday, March 14, 2025 at 12:17:25 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 15

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

\[ x \left (t \right ) = t \left (c_{1} {\mathrm e}^{t}+c_{3} t +c_{2} \right ) \]

✓ Mathematica. Time used: 0.18 (sec). Leaf size: 208

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

\[ x(t)\to t^{3/2} \exp \left (\frac {1}{2} \left (2 \int _1^t\frac {K[1]-1}{2 K[1]}dK[1]+t+3\right )\right ) \left (\int _1^t\exp \left (-2 \int _1^{K[2]}\frac {K[1]-1}{2 K[1]}dK[1]\right )dK[2] \left (\int _1^t\frac {\exp \left (-\frac {K[4]}{2}+\int _1^{K[4]}\frac {K[1]-1}{2 K[1]}dK[1]-\frac {3}{2}\right ) c_1}{K[4]^{3/2}}dK[4]+c_3\right )+\int _1^t-\frac {\exp \left (-\frac {K[3]}{2}+\int _1^{K[3]}\frac {K[1]-1}{2 K[1]}dK[1]-\frac {3}{2}\right ) c_1 \int _1^{K[3]}\exp \left (-2 \int _1^{K[2]}\frac {K[1]-1}{2 K[1]}dK[1]\right )dK[2]}{K[3]^{3/2}}dK[3]+c_2\right ) \]

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(t**3*Derivative(x(t), (t, 3)) - t**2*(t + 3)*Derivative(x(t), (t, 2)) + 2*t*(t + 3)*Derivative(x(t), t) - (2*t + 6)*x(t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

NotImplementedError : The given ODE Derivative(x(t), t) - (t**3*Derivative(x(t), (t, 2)) - t**3*Derivative(x(t), (t, 3)) + 3*t**2*Derivative(x(t), (t, 2)) + 2*t*x(t) + 6*x(t))/(2*t*(t + 3)) cannot be solved by the factorable group method