14.17.6 problem 24

Internal problem ID [2684]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.10, Some useful properties of Laplace transform. Excercises page 238
Problem number : 24
Date solved : Tuesday, March 04, 2025 at 02:34:03 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\left \{\begin {array}{cc} 2 & 0\le t \le 3 \\ 3 t -7 & 3<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: 1.424 (sec). Leaf size: 30

ode:=diff(diff(y(t),t),t)+y(t) = piecewise(0 <= t and t <= 3,2,3 < t,3*t-7); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = -2 \cos \left (t \right )-\left (\left \{\begin {array}{cc} -2 & t <3 \\ 7+3 \sin \left (t -3\right )-3 t & 3\le t \end {array}\right .\right ) \]

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 42

ode=D[y[t],{t,2}]+y[t]==Piecewise[{ {2,0<=t<=3},{3*t-7,3<t}}]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 0 & t\leq 0 \\ 2-2 \cos (t) & 0<t\leq 3 \\ 3 t-2 \cos (t)+3 \sin (3-t)-7 & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 0.387 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((2, (t >= 0) & (t <= 3)), (3*t - 7, t > 3)) + 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)
 

\[ y{\left (t \right )} = \begin {cases} 2 & \text {for}\: t \geq 0 \wedge t \leq 3 \\3 t - 7 & \text {for}\: t > 3 \\\text {NaN} & \text {otherwise} \end {cases} - 2 \cos {\left (t \right )} \]