68.11.49 problem 61

Internal problem ID [17586]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.3, page 156
Problem number : 61
Date solved : Thursday, October 02, 2025 at 02:25:52 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\left \{\begin {array}{cc} 0 & 0\le t <\pi \\ 10 & \pi \le t <2 \pi \\ 0 & 2 \pi \le t \end {array}\right . \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.107 (sec). Leaf size: 33

ode:=diff(diff(y(t),t),t)+4*y(t) = piecewise(0 <= t and t < Pi,0,Pi <= t and t < 2*Pi,10,2*Pi <= t,0); 
ic:=[y(0) = 0, D(y)(0) = 2]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \sin \left (2 t \right )+\left (\left \{\begin {array}{cc} 0 & t <\pi \\ \frac {5}{2}-\frac {5 \cos \left (2 t \right )}{2} & t <2 \pi \\ 0 & 2 \pi \le t \end {array}\right .\right ) \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 37

ode=D[y[t],{t,2}]+4*y[t]==Piecewise[{ {0,0<=t<Pi},{10,Pi<=t<2*Pi},{0,t>=2*Pi} }]; 
ic={y[0]==0,Derivative[1][y][0] ==2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} \sin (2 t) & t>2 \pi \lor t\leq \pi \\ -\frac {5}{2} \cos (2 t)+\sin (2 t)+\frac {5}{2} & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.315 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((0, (t >= 0) & (t < pi)), (10, (t >= pi) & (t < 2*pi)), (0, t >= 2*pi)) + 4*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \begin {cases} 0 & \text {for}\: t \geq 0 \wedge t < \pi \\\frac {5}{2} & \text {for}\: t \geq \pi \wedge t < 2 \pi \\0 & \text {for}\: t \geq 2 \pi \\\text {NaN} & \text {otherwise} \end {cases} + \sin {\left (2 t \right )} \]