57.14.7 problem 2

Internal problem ID [14464]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.5 Higher order equations. Exercises page 130
Problem number : 2
Date solved : Thursday, October 02, 2025 at 09:37:32 AM
CAS classification : [[_3rd_order, _missing_x]]

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

With initial conditions

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

✓ Maple. Time used: 0.476 (sec). Leaf size: 248

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

\[ x = \frac {8 \,{\mathrm e}^{-\frac {\left (4+\frac {\left (388+36 \sqrt {113}\right )^{{2}/{3}}}{4}+\left (388+36 \sqrt {113}\right )^{{1}/{3}}\right ) t}{3 \left (388+36 \sqrt {113}\right )^{{1}/{3}}}} \left (\left (\frac {1017}{4}+\left (\sqrt {113}+11\right ) \left (388+36 \sqrt {113}\right )^{{1}/{3}}+\frac {\left (-\sqrt {113}-25\right ) \left (388+36 \sqrt {113}\right )^{{2}/{3}}}{32}+\frac {97 \sqrt {113}}{4}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (388+36 \sqrt {113}\right )^{{2}/{3}}-16\right ) t}{12 \left (388+36 \sqrt {113}\right )^{{1}/{3}}}\right )+\sqrt {3}\, \left (\left (\sqrt {113}+11\right ) \left (388+36 \sqrt {113}\right )^{{1}/{3}}+\frac {\left (\sqrt {113}+25\right ) \left (388+36 \sqrt {113}\right )^{{2}/{3}}}{32}\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (388+36 \sqrt {113}\right )^{{2}/{3}}-16\right ) t}{12 \left (388+36 \sqrt {113}\right )^{{1}/{3}}}\right )-\left (-\frac {1017}{8}+\left (\sqrt {113}+11\right ) \left (388+36 \sqrt {113}\right )^{{1}/{3}}+\frac {\left (-\sqrt {113}-25\right ) \left (388+36 \sqrt {113}\right )^{{2}/{3}}}{32}-\frac {97 \sqrt {113}}{8}\right ) {\mathrm e}^{\frac {t \left (16+\left (388+36 \sqrt {113}\right )^{{2}/{3}}\right )}{4 \left (388+36 \sqrt {113}\right )^{{1}/{3}}}}\right )}{3 \left (97 \sqrt {113}+1017\right )} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 748

ode=D[x[t],{t,3}]+D[x[t],{t,2}]-D[x[t],t]-4*x[t]==0; 
ic={x[0]==1,Derivative[1][x][0 ]==0,Derivative[2][x][0]==-1}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-4*x(t) - Derivative(x(t), t) + Derivative(x(t), (t, 2)) + Derivative(x(t), (t, 3)),0) 
ics = {x(0): 1, Subs(Derivative(x(t), t), t, 0): 0, Subs(Derivative(x(t), (t, 2)), t, 0): -1} 
dsolve(ode,func=x(t),ics=ics)
 

Timed Out