40.7.4 problem 21

Internal problem ID [8651]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 6. Laplace Transforms. Problem set 6.3, page 224
Problem number : 21
Date solved : Tuesday, September 30, 2025 at 05:40:12 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+9 y&=\left \{\begin {array}{cc} 8 \sin \left (t \right ) & 0<t <\pi \\ 0 & \pi <t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=4 \\ \end{align*}

✓ Maple. Time used: 0.302 (sec). Leaf size: 26

ode:=diff(diff(y(t),t),t)+9*y(t) = piecewise(0 < t and t < Pi,8*sin(t),Pi < t,0); 
ic:=[y(0) = 0, D(y)(0) = 4]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = 4 \left (\left \{\begin {array}{cc} \sin \left (t \right ) \cos \left (t \right )^{2} & t <\pi \\ \frac {\sin \left (3 t \right )}{3} & \pi \le t \end {array}\right .\right ) \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 30

ode=D[y[t],{t,2}]+9*y[t]==Piecewise[{{8*Sin[t],0<t<Pi},{0,t>Pi}}]; 
ic={y[0]==0,Derivative[1][y][0] ==4}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {4}{3} \sin (3 t) & t>\pi \lor t\leq 0 \\ \sin (t)+\sin (3 t) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.278 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((8*sin(t), (t <= pi) & (t > 0)), (0, t > pi)) + 9*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 4} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{2} \cos {\left (3 t \right )} + \begin {cases} \sin {\left (t \right )} & \text {for}\: t \leq \pi \wedge t > 0 \\0 & \text {for}\: t > \pi \\\text {NaN} & \text {otherwise} \end {cases} + \frac {4 \sin {\left (3 t \right )}}{3} \]