65.9.4 problem 16.1 (iv)

Internal problem ID [13704]
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 (iv)
Date solved : Wednesday, March 05, 2025 at 10:13:20 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} x^{\prime \prime \prime \prime }-5 x^{\prime \prime }+4 x&={\mathrm e}^{t} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 36

ode:=diff(diff(diff(diff(x(t),t),t),t),t)-5*diff(diff(x(t),t),t)+4*x(t) = exp(t); 
dsolve(ode,x(t), singsol=all);
 

\[ x \left (t \right ) = -\frac {{\mathrm e}^{-2 t} \left (\left (t -6 c_{1} \right ) {\mathrm e}^{3 t}-6 c_{3} {\mathrm e}^{t}-6 c_4 \,{\mathrm e}^{4 t}-6 c_{2} \right )}{6} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 45

ode=D[x[t],{t,4}]-5*D[x[t],{t,2}]+4*x[t]==Exp[t]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to e^{-2 t} \left (c_2 e^t+e^{3 t} \left (-\frac {t}{6}-\frac {1}{36}+c_3\right )+c_4 e^{4 t}+c_1\right ) \]

✓ Sympy. Time used: 0.156 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(4*x(t) - exp(t) - 5*Derivative(x(t), (t, 2)) + Derivative(x(t), (t, 4)),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = C_{2} e^{- 2 t} + C_{3} e^{- t} + C_{4} e^{2 t} + \left (C_{1} - \frac {t}{6}\right ) e^{t} \]