76.18.12 problem 24

Internal problem ID [17652]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.2 (Properties of the Laplace transform). Problems at page 309
Problem number : 24
Date solved : Monday, March 31, 2025 at 04:23:41 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\left \{\begin {array}{cc} t & 0\le t <1 \\ 0 & 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: 0.382 (sec). Leaf size: 26

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

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

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 44

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

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} t-\sin (t) & 0<t\leq 1 \\ \cos (1-t)-\sin (1-t)-\sin (t) & t>1 \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 0.311 (sec). Leaf size: 10

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

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