28.3.5 problem 6.40

Internal problem ID [4518]
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 : 6.40
Date solved : Tuesday, March 04, 2025 at 06:50:56 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+y&=4 \,{\mathrm e}^{-t}+2 \,{\mathrm e}^{t} \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: 3.041 (sec). Leaf size: 20

ode:=diff(diff(y(t),t),t)-2*diff(y(t),t)+y(t) = 4*exp(-t)+2*exp(t); 
ic:=y(0) = -1, D(y)(0) = 2; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = {\mathrm e}^{t} \left (t^{2}+5 t -2\right )+{\mathrm e}^{-t} \]

✓ Mathematica. Time used: 0.743 (sec). Leaf size: 69

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

\[ y(t)\to \frac {1}{3} e^{-t} \left (2 \left (e^{2 t}+6\right )+5 \sqrt {3} e^{t/2} \sin \left (\frac {\sqrt {3} t}{2}\right )-17 e^{t/2} \cos \left (\frac {\sqrt {3} t}{2}\right )\right ) \]

✓ Sympy. Time used: 0.243 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t) - 2*exp(t) - 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 4*exp(-t),0) 
ics = {y(0): -1, Subs(Derivative(y(t), t), t, 0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (t \left (t + 5\right ) - 2\right ) e^{t} + e^{- t} \]