31.1.4 problem 10.2.8 part(2)

Internal problem ID [7679]
Book : Basic Training in Mathematics. By R. Shankar. Plenum Press. NY. 1995
Section : Chapter 10, Differential equations. Section 10.2, ODEs with constant Coefficients. page 307
Problem number : 10.2.8 part(2)
Date solved : Tuesday, September 30, 2025 at 04:55:51 PM
CAS classification : [[_high_order, _missing_x]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 53

ode:=diff(diff(diff(diff(x(t),t),t),t),t)+x(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

\[ x = -\left (\left (-c_4 \,{\mathrm e}^{\sqrt {2}\, t}-c_3 \right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right )+\sin \left (\frac {\sqrt {2}\, t}{2}\right ) \left (c_2 \,{\mathrm e}^{\sqrt {2}\, t}+c_1 \right )\right ) {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 65

ode=D[x[t],{t,4}]+x[t]==0; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{-\frac {t}{\sqrt {2}}} \left (\left (c_1 e^{\sqrt {2} t}+c_2\right ) \cos \left (\frac {t}{\sqrt {2}}\right )+\left (c_4 e^{\sqrt {2} t}+c_3\right ) \sin \left (\frac {t}{\sqrt {2}}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.091 (sec). Leaf size: 70

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

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