28.3.13 problem 6.48

Internal problem ID [4526]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 6. The Laplace Transform and Its Applications. Problems at page 291
Problem number : 6.48
Date solved : Tuesday, March 04, 2025 at 06:51:06 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-3 y^{\prime }+2 y&={\mathrm e}^{t} \operatorname {Heaviside}\left (t -2\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Maple. Time used: 3.495 (sec). Leaf size: 31

ode:=diff(diff(y(t),t),t)-3*diff(y(t),t)+2*y(t) = exp(t)*Heaviside(t-2); 
ic:=y(0) = 1, D(y)(0) = 2; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-2+2 t}-{\mathrm e}^{t} \left (-1+t \right ) \operatorname {Heaviside}\left (t -2\right )+{\mathrm e}^{2 t} \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 33

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

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

✓ Sympy. Time used: 0.576 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(2*y(t) - exp(t)*Heaviside(t - 2) - 3*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (\frac {\theta \left (t - 2\right )}{e^{2}} + 1\right ) e^{t} - \begin {cases} 0 & \text {for}\: \left |{t}\right | < 2 \\t {G_{2, 2}^{0, 2}\left (\begin {matrix} 0, 1 & \\ & -1, 0 \end {matrix} \middle | {\frac {t}{2}} \right )} & \text {otherwise} \end {cases} - \theta \left (t - 2\right )\right ) e^{t} \]