5.3.11 problem 18

Internal problem ID [1493]
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 : 18
Date solved : Tuesday, September 30, 2025 at 04:34:38 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.253 (sec). Leaf size: 27

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

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

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 39

ode=D[y[t],{t,2}]+4*y[t]==Piecewise[{{1,0<t<1},{0,1<=t<Infinity}}]; 
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 {\sin ^2(t)}{2} & 0<t\leq 1 \\ -\frac {1}{2} \sin (1) \sin (1-2 t) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.299 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((1, (t >= 0) & (t < 1)), (0, (t >= 1) & (t < oo))) + 4*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)
 

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