5.3.10 problem 17

Internal problem ID [1492]
Book : Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section : Chapter 6.2, The Laplace Transform. Solution of Initial Value Problems. page 255
Problem number : 17
Date solved : Tuesday, September 30, 2025 at 04:34:37 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\left \{\begin {array}{cc} 1 & 0\le t <\pi \\ 0 & \pi \le t <\infty \end {array}\right . \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.283 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)+4*y(t) = piecewise(0 <= t and t < Pi,1,Pi <= t and t < infinity,0); 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \left \{\begin {array}{cc} \frac {3 \cos \left (2 t \right )}{4}+\frac {1}{4} & t <\pi \\ \cos \left (2 t \right ) & \pi \le t \end {array}\right . \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 31

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

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} \cos (2 t) & t>\pi \lor t\leq 0 \\ \frac {1}{4} (3 \cos (2 t)+1) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.337 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((1, (t > 0) & (t < pi)), (0, (t >= pi) & (t < oo))) + 4*y(t) + Derivative(y(t), (t, 2)) - 1,0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{2} \cos {\left (2 t \right )} + \begin {cases} \frac {1}{2} & \text {for}\: t > 0 \wedge t < \pi \\\frac {1}{4} & \text {for}\: t \geq \pi \wedge t < \infty \\\text {NaN} & \text {otherwise} \end {cases} \]