84.37.11 problem 26.11

Internal problem ID [22353]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 26. Solutions of linear differential equations with constant coefficients by Laplace transform. Solved problems. Page 159
Problem number : 26.11
Date solved : Thursday, October 02, 2025 at 08:37:52 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\left \{\begin {array}{cc} 0 & t <1 \\ 2 & 1\le t \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: 0.097 (sec). Leaf size: 20

ode:=diff(diff(y(t),t),t)+y(t) = piecewise(t < 1,0,1 <= t,2); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \left \{\begin {array}{cc} 0 & t <1 \\ 2-2 \cos \left (t -1\right ) & 1\le t \end {array}\right . \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 26

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

✓ Sympy. Time used: 0.134 (sec). Leaf size: 8

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((1, t < 1), (2, True)) + 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)
 

\[ y{\left (t \right )} = \begin {cases} 1 & \text {for}\: t < 1 \\2 & \text {otherwise} \end {cases} - \cos {\left (t \right )} \]