63.14.7 problem 2
Internal
problem
ID
[13104]
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
:
Wednesday, March 05, 2025 at 09:17:23 PM
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: 0.549 (sec). Leaf size: 261
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,ic],x(t), singsol=all);
\[
x \left (t \right ) = \frac {\left (\left (\left (32 \sqrt {113}+352\right ) \left (388+36 \sqrt {113}\right )^{{1}/{3}}+\left (-\sqrt {113}-25\right ) \left (388+36 \sqrt {113}\right )^{{2}/{3}}+776 \sqrt {113}+8136\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (388+36 \sqrt {113}\right )^{{2}/{3}}-16\right ) t}{12 \left (388+36 \sqrt {113}\right )^{{1}/{3}}}\right )+32 \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (388+36 \sqrt {113}\right )^{{2}/{3}}-16\right ) t}{12 \left (388+36 \sqrt {113}\right )^{{1}/{3}}}\right ) \left (\left (\sqrt {113}+11\right ) \left (388+36 \sqrt {113}\right )^{{1}/{3}}+\frac {\left (388+36 \sqrt {113}\right )^{{2}/{3}} \left (\sqrt {113}+25\right )}{32}\right )\right ) {\mathrm e}^{-\frac {t \left (4+\frac {\left (388+36 \sqrt {113}\right )^{{2}/{3}}}{4}+\left (388+36 \sqrt {113}\right )^{{1}/{3}}\right )}{3 \left (388+36 \sqrt {113}\right )^{{1}/{3}}}}-32 \left (\left (\sqrt {113}+11\right ) \left (388+36 \sqrt {113}\right )^{{1}/{3}}+\left (-\frac {\sqrt {113}}{32}-\frac {25}{32}\right ) \left (388+36 \sqrt {113}\right )^{{2}/{3}}-\frac {97 \sqrt {113}}{8}-\frac {1017}{8}\right ) {\mathrm e}^{-\frac {\left (-\frac {\left (388+36 \sqrt {113}\right )^{{2}/{3}}}{2}+\left (388+36 \sqrt {113}\right )^{{1}/{3}}-8\right ) t}{3 \left (388+36 \sqrt {113}\right )^{{1}/{3}}}}}{1164 \sqrt {113}+12204}
\]
✓ Mathematica. Time used: 0.01 (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