80.6.15 problem 15

Internal problem ID [21305]
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 : 15
Date solved : Thursday, October 02, 2025 at 07:28:16 PM
CAS classification : [[_high_order, _missing_x]]

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

With initial conditions

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

✓ Maple. Time used: 0.405 (sec). Leaf size: 64

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

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

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 73

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

\begin{align*} x(t)&\to -\frac {e^{-\sqrt {2-\sqrt {3}} t}-e^{-\sqrt {2+\sqrt {3}} t}}{\sqrt {2-\sqrt {3}}-\sqrt {2+\sqrt {3}}} \end{align*}

✓ Sympy. Time used: 0.174 (sec). Leaf size: 131

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

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