68.21.4 problem 18

Internal problem ID [17929]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 5. Applications of Higher Order Equations. Exercises 5.3, page 249
Problem number : 18
Date solved : Thursday, October 02, 2025 at 02:29:58 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+4 x^{\prime }+13 x&=\left \{\begin {array}{cc} 1 & 0\le t <\pi \\ 1-t & \pi \le t <2 \pi \\ 0 & 2 \pi \le t \end {array}\right . \end{align*}

With initial conditions

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

✓ Maple. Time used: 1.848 (sec). Leaf size: 165

ode:=diff(diff(x(t),t),t)+4*diff(x(t),t)+13*x(t) = piecewise(0 <= t and t < Pi,1,Pi <= t and t < 2*Pi,1-t,2*Pi <= t,0); 
ic:=[x(0) = 0, D(x)(0) = 0]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {\left (\left \{\begin {array}{cc} 0 & t <0 \\ 3+\left (-2 \sin \left (3 t \right )-3 \cos \left (3 t \right )\right ) {\mathrm e}^{-2 t} & t <\pi \\ \frac {\left (\left (-39 \pi +12\right ) \cos \left (3 t \right )+\left (-26 \pi -5\right ) \sin \left (3 t \right )\right ) {\mathrm e}^{2 \pi -2 t}}{13}-3 \,{\mathrm e}^{-2 t} \cos \left (3 t \right )-2 \,{\mathrm e}^{-2 t} \sin \left (3 t \right )-3 t +\frac {51}{13} & t <2 \pi \\ -3 \left (\left (\left (\pi -\frac {4}{13}\right ) \cos \left (3 t \right )+\frac {2 \left (\pi +\frac {5}{26}\right ) \sin \left (3 t \right )}{3}\right ) {\mathrm e}^{2 \pi }+\left (\left (2 \pi -\frac {17}{13}\right ) \cos \left (3 t \right )+\frac {4 \left (\pi -\frac {21}{52}\right ) \sin \left (3 t \right )}{3}\right ) {\mathrm e}^{4 \pi }+\cos \left (3 t \right )+\frac {2 \sin \left (3 t \right )}{3}\right ) {\mathrm e}^{-2 t} & 2 \pi \le t \end {array}\right .\right )}{39} \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 60

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

\begin{align*} x(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} 0 & t\leq 0 \\ 1-\cos (t) & 0<t\leq \pi \\ -t-(1+\pi ) \cos (t)-\sin (t)+1 & \pi <t\leq 2 \pi \\ -3 \pi \cos (t)-2 \sin (t) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.343 (sec). Leaf size: 17

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

\[ x{\left (t \right )} = \begin {cases} 1 & \text {for}\: t \geq 0 \wedge t < \pi \\1 - t & \text {for}\: t \geq \pi \wedge t \leq 2 \pi \\0 & \text {for}\: t > 2 \pi \\\text {NaN} & \text {otherwise} \end {cases} - \cos {\left (t \right )} \]