74.21.2 problem 16

Internal problem ID [16569]
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 : 16
Date solved : Monday, March 31, 2025 at 02:59:01 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+x&=\left \{\begin {array}{cc} \cos \left (t \right ) & 0\le t <\pi \\ 0 & \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: 0.124 (sec). Leaf size: 21

ode:=diff(diff(x(t),t),t)+x(t) = piecewise(0 <= t and t < Pi,cos(t),Pi <= t,0); 
ic:=x(0) = 0, D(x)(0) = 0; 
dsolve([ode,ic],x(t), singsol=all);
 

\[ x = \frac {\sin \left (t \right ) \left (\left \{\begin {array}{cc} 0 & t <0 \\ t & t <\pi \\ \pi & \pi \le t \end {array}\right .\right )}{2} \]

✓ Mathematica. Time used: 13.333 (sec). Leaf size: 83

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

\[ x(t)\to \sin (t) \left (\int _1^t\cos (K[1]) \left ( \begin {array}{cc} \{ & \begin {array}{cc} \cos (K[1]) & 0\leq K[1]<\pi \\ 0 & \text {True} \\ \end {array} \\ \end {array} \right )dK[1]-\int _1^0\cos (K[1]) \left ( \begin {array}{cc} \{ & \begin {array}{cc} \cos (K[1]) & 0\leq K[1]<\pi \\ 0 & \text {True} \\ \end {array} \\ \end {array} \right )dK[1]\right )+\cos (t) \left ( \begin {array}{cc} \{ & \begin {array}{cc} -\frac {1}{2} \sin ^2(t) & 0<t\leq \pi \\ 0 & \text {True} \\ \end {array} \\ \end {array} \right ) \]

✓ Sympy. Time used: 0.345 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-Piecewise((cos(t), (t >= 0) & (t < pi)), (0, t >= 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)
 

\[ x{\left (t \right )} = \begin {cases} \frac {t \sin {\left (t \right )}}{2} + \frac {\cos {\left (t \right )}}{4} & \text {for}\: t \geq 0 \wedge t < \pi \\0 & \text {for}\: t \geq \pi \\\text {NaN} & \text {otherwise} \end {cases} - \frac {\cos {\left (t \right )}}{4} \]