65.9.3 problem 16.1 (iii)

Internal problem ID [13703]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 16, Higher order linear equations with constant coefficients. Exercises page 153
Problem number : 16.1 (iii)
Date solved : Wednesday, March 05, 2025 at 10:13:19 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime \prime \prime }-4 x^{\prime \prime \prime }+8 x^{\prime \prime }-8 x^{\prime }+4 x&=\sin \left (t \right ) \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 33

ode:=diff(diff(diff(diff(x(t),t),t),t),t)-4*diff(diff(diff(x(t),t),t),t)+8*diff(diff(x(t),t),t)-8*diff(x(t),t)+4*x(t) = sin(t); 
dsolve(ode,x(t), singsol=all);
 

\[ x \left (t \right ) = \left (\left (c_4 t +c_{1} \right ) \cos \left (t \right )+\sin \left (t \right ) \left (c_{3} t +c_{2} \right )\right ) {\mathrm e}^{t}+\frac {4 \cos \left (t \right )}{25}-\frac {3 \sin \left (t \right )}{25} \]

✓ Mathematica. Time used: 0.136 (sec). Leaf size: 152

ode=D[x[t],{t,4}]-4*D[x[t],{t,3}]+8*D[x[t],{t,2}]-8*D[x[t],t]+4*x[t]==Sin[t]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to e^t \left (t \sin (t) \int _1^t-\frac {1}{2} e^{-K[2]} \sin ^2(K[2])dK[2]+\cos (t) \int _1^t\frac {1}{2} e^{-K[3]} (\cos (K[3]) K[3]-\sin (K[3])) \sin (K[3])dK[3]+t \cos (t) \int _1^t-\frac {1}{4} e^{-K[4]} \sin (2 K[4])dK[4]+\sin (t) \int _1^t\frac {1}{2} e^{-K[1]} \sin (K[1]) (\cos (K[1])+K[1] \sin (K[1]))dK[1]+c_3 \cos (t)+c_4 t \cos (t)+c_1 \sin (t)+c_2 t \sin (t)\right ) \]

✓ Sympy. Time used: 0.282 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(4*x(t) - sin(t) - 8*Derivative(x(t), t) + 8*Derivative(x(t), (t, 2)) - 4*Derivative(x(t), (t, 3)) + Derivative(x(t), (t, 4)),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (\left (C_{1} + C_{2} t\right ) \sin {\left (t \right )} + \left (C_{3} + C_{4} t\right ) \cos {\left (t \right )}\right ) e^{t} - \frac {3 \sin {\left (t \right )}}{25} + \frac {4 \cos {\left (t \right )}}{25} \]