46.7.5 problem 22

Internal problem ID [7366]
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 : 22
Date solved : Sunday, March 30, 2025 at 11:55:23 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=\left \{\begin {array}{cc} 4 t & 0<t <1 \\ 8 & 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: 0.229 (sec). Leaf size: 71

ode:=diff(diff(y(t),t),t)+3*diff(y(t),t)+2*y(t) = piecewise(0 < t and t < 1,4*t,1 < t,8); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \left \{\begin {array}{cc} -3-{\mathrm e}^{-2 t}+4 \,{\mathrm e}^{-t}+2 t & t <1 \\ 1-{\mathrm e}^{-2}+4 \,{\mathrm e}^{-1} & t =1 \\ 4-{\mathrm e}^{-2 t}+4 \,{\mathrm e}^{-t}+3 \,{\mathrm e}^{2-2 t}-8 \,{\mathrm e}^{1-t} & 1<t \end {array}\right . \]

✓ Mathematica. Time used: 0.047 (sec). Leaf size: 70

ode=D[y[t],{t,2}]+3*D[y[t],t]+2*y[t]==Piecewise[{{4*t,0<t<1},{8,t>1}}]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 0 & t\leq 0 \\ 2 t-e^{-2 t}+4 e^{-t}-3 & 0<t\leq 1 \\ e^{-2 t} \left (-1+3 e^2+4 e^t+4 e^{2 t}-8 e^{t+1}\right ) & \text {True} \\ \end {array} \\ \end {array} \]

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((4*t, (t <= 1) & (t > 0)), (8, 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)