76.20.5 problem 5

Internal problem ID [17682]
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 5.6 (Differential equations with Discontinuous Forcing Functions). Problems at page 342
Problem number : 5
Date solved : Monday, March 31, 2025 at 04:24:25 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=\left \{\begin {array}{cc} 1 & 0\le t <10 \\ 0 & 10\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: 0.165 (sec). Leaf size: 52

ode:=diff(diff(y(t),t),t)+3*diff(y(t),t)+2*y(t) = piecewise(0 <= t and t < 10,1,10 <= t,0); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {\left (\left \{\begin {array}{cc} \left ({\mathrm e}^{-t}-1\right )^{2} & t <10 \\ -2 \,{\mathrm e}^{-10}+{\mathrm e}^{-20}+2 & t =10 \\ -\left ({\mathrm e}^{20}-2 \,{\mathrm e}^{10+t}+2 \,{\mathrm e}^{t}-1\right ) {\mathrm e}^{-2 t} & 10<t \end {array}\right .\right )}{2} \]

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 61

ode=D[y[t],{t,2}]+3*D[y[t],t]+2*y[t]==Piecewise[{  {1,0<= t <10}, {0,t>=10}}]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 0 & t\leq 0 \\ \frac {1}{2} e^{-2 t} \left (-1+e^t\right )^2 & 0<t\leq 10 \\ \frac {1}{2} e^{-2 t} \left (-1+e^{10}\right ) \left (-1-e^{10}+2 e^t\right ) & \text {True} \\ \end {array} \\ \end {array} \]

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((1, (t >= 0) & (t < 10)), (0, t >= 10)) + 2*y(t) + 3*Derivative(y(t), 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)