68.11.48 problem 60

Internal problem ID [17585]
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 : 60
Date solved : Thursday, October 02, 2025 at 02:25:47 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+9 \pi ^{2} y&=\left \{\begin {array}{cc} 2 t & 0\le t <\pi \\ 2 t -2 \pi & \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 )&=0 \\ \end{align*}

✓ Maple. Time used: 1.180 (sec). Leaf size: 121

ode:=diff(diff(y(t),t),t)+9*Pi^2*y(t) = piecewise(0 <= t and t < Pi,2*t,Pi <= t and t < 2*Pi,2*t-2*Pi,2*Pi <= t,0); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {2 \left (\left \{\begin {array}{cc} 0 & t <0 \\ 3 \pi t -\sin \left (3 \pi t \right ) & t <\pi \\ 3 \cos \left (3 \pi ^{2}-3 \pi t \right ) \pi ^{2}-3 \pi ^{2}+3 \pi t -\sin \left (3 \pi t \right ) & t <2 \pi \\ 3 \left (\cos \left (3 \pi ^{2}-3 \pi t \right )+\cos \left (6 \pi ^{2}-3 \pi t \right )\right ) \pi ^{2}-\sin \left (3 \pi t \right )-\sin \left (6 \pi ^{2}-3 \pi t \right ) & 2 \pi \le t \end {array}\right .\right )}{27 \pi ^{3}} \]

✓ Mathematica. Time used: 0.048 (sec). Leaf size: 146

ode=D[y[t],{t,2}]+9*Pi^2*y[t]==Piecewise[{ {2*t,0<=t<Pi},{2*(t-Pi),Pi<=t<2*Pi},{0,t>=2*Pi} }]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {2 (3 \pi t-\sin (3 \pi t))}{27 \pi ^3} & 0<t\leq \pi \\ \frac {2 \left (3 \pi (t-\pi )+3 \pi ^2 \cos (3 \pi (\pi -t))-\sin (3 \pi t)\right )}{27 \pi ^3} & \pi <t\leq 2 \pi \\ \frac {4 \cos \left (\frac {3 \pi ^2}{2}\right ) \left (3 \pi ^2 \cos \left (\frac {3}{2} \pi (3 \pi -2 t)\right )+\sin \left (\frac {3}{2} \pi (\pi -2 t)\right )-\sin \left (\frac {3}{2} \pi (3 \pi -2 t)\right )\right )}{27 \pi ^3} & t>2 \pi \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.567 (sec). Leaf size: 78

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((2*t, (t >= 0) & (t < pi)), (2*t - 2*pi, (t >= pi) & (t < 2*pi)), (0, t >= 2*pi)) + 9*pi**2*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \begin {cases} \frac {2 t}{9 \pi ^{2}} & \text {for}\: t \geq 0 \wedge t < \pi \\- \frac {t \cos {\left (6 \pi t \right )}}{9 \pi ^{2}} + \frac {t}{9 \pi ^{2}} + \frac {\sin {\left (6 \pi t \right )}}{27 \pi ^{3}} & \text {for}\: t \geq 2 \pi \wedge t < \pi \\\frac {2 t}{9 \pi ^{2}} - \frac {2}{9 \pi } & \text {for}\: t \geq \pi \wedge t < 2 \pi \\0 & \text {for}\: t \geq 2 \pi \\\text {NaN} & \text {otherwise} \end {cases} - \frac {2 \sin {\left (3 \pi t \right )}}{27 \pi ^{3}} \]