80.6.17 problem 17

Internal problem ID [21307]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 6. Higher order linear equations. Excercise 6.5 at page 133
Problem number : 17
Date solved : Thursday, October 02, 2025 at 07:28:17 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} x^{\prime \prime \prime \prime }+2 x^{\prime \prime }-4 x&=0 \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=1 \\ x^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.121 (sec). Leaf size: 100

ode:=diff(diff(diff(diff(x(t),t),t),t),t)+2*diff(diff(x(t),t),t)-4*x(t) = 0; 
ic:=[x(0) = 1, D(x)(0) = 1]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {-\sqrt {\sqrt {5}+1}\, \left (c_4 -1\right ) {\mathrm e}^{-\sqrt {\sqrt {5}-1}\, t}+2 \left (\frac {1}{2}+\left (-c_2 -\frac {c_4}{2}+\frac {1}{2}\right ) \sqrt {\sqrt {5}-1}\right ) \sin \left (\sqrt {\sqrt {5}+1}\, t \right )+2 \sqrt {\sqrt {5}+1}\, \left (c_2 \sinh \left (\sqrt {\sqrt {5}-1}\, t \right )+\frac {c_4 \cos \left (\sqrt {\sqrt {5}+1}\, t \right )}{2}\right )}{\sqrt {\sqrt {5}+1}} \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 133

ode=D[x[t],{t,4}]+2*D[x[t],{t,2}]-4*x[t]==0; 
ic={x[0]==1,Derivative[1][x][0] ==1}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{-\sqrt {\sqrt {5}-1} t} \left (c_3 e^{2 \sqrt {\sqrt {5}-1} t}-\frac {1}{\sqrt {\sqrt {5}-1}}+\frac {2 c_2}{\sqrt {5}-1}+c_3\right )+\left (\frac {1}{\sqrt {\sqrt {5}-1}}+1-\frac {2 c_2}{\sqrt {5}-1}-2 c_3\right ) \cos \left (\sqrt {1+\sqrt {5}} t\right )+c_2 \sin \left (\sqrt {1+\sqrt {5}} t\right ) \end{align*}

✓ Sympy. Time used: 0.134 (sec). Leaf size: 156

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-4*x(t) + 2*Derivative(x(t), (t, 2)) + Derivative(x(t), (t, 4)),0) 
ics = {x(0): 1, Subs(Derivative(x(t), t), t, 0): 1} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = C_{3} \sin {\left (t \sqrt {1 + \sqrt {5}} \right )} + C_{4} \cos {\left (t \sqrt {1 + \sqrt {5}} \right )} + \left (C_{3} \left (\frac {1}{4} + \frac {\sqrt {5}}{4}\right ) - \frac {C_{4}}{2} - \frac {\sqrt {5} \sqrt {-1 + \sqrt {5}}}{8} - \frac {\sqrt {-1 + \sqrt {5}}}{8} + \frac {1}{2}\right ) e^{- t \sqrt {-1 + \sqrt {5}}} + \left (C_{3} \left (- \frac {\sqrt {5}}{4} - \frac {1}{4}\right ) - \frac {C_{4}}{2} + \frac {\sqrt {-1 + \sqrt {5}}}{8} + \frac {\sqrt {5} \sqrt {-1 + \sqrt {5}}}{8} + \frac {1}{2}\right ) e^{t \sqrt {-1 + \sqrt {5}}} \]