40.7.7 problem 24

Internal problem ID [8654]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 6. Laplace Transforms. Problem set 6.3, page 224
Problem number : 24
Date solved : Tuesday, September 30, 2025 at 05:40:15 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=\left \{\begin {array}{cc} 1 & 0<t <1 \\ 0 & 1<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: 0.154 (sec). Leaf size: 52

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

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

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 57

ode=D[y[t],{t,2}]+3*D[y[t],t]+2*y[t]==Piecewise[{{1,0<t<1},{0,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} 0 & t\leq 0 \\ \frac {1}{2} e^{-2 t} \left (-1+e^t\right )^2 & 0<t\leq 1 \\ \frac {1}{2} (-1+e) e^{-2 t} \left (-1-e+2 e^t\right ) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✗ Sympy

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