68.11.47 problem 59

Internal problem ID [17584]
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 : 59
Date solved : Thursday, October 02, 2025 at 02:25:46 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+9 y&=\left \{\begin {array}{cc} 2 t & 0\le t <\pi \\ 0 & \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: 0.228 (sec). Leaf size: 41

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

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

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 49

ode=D[y[t],{t,2}]+9*y[t]==Piecewise[{ {2*t,0<=t<Pi},{0,t>=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} 0 & t\leq 0 \\ -\frac {2}{27} (\sin (3 t)-3 t) & 0<t\leq \pi \\ -\frac {2}{27} (3 \pi \cos (3 t)+2 \sin (3 t)) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.232 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((2*t, (t >= 0) & (t < pi)), (0, t >= pi)) + 9*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)
 

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