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 : Monday, January 27, 2025 at 09:22:37 AM
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*}

✓ Solution by Maple

Time used: 3.041 (sec). Leaf size: 20

dsolve([diff(y(t),t$2)-2*diff(y(t),t)+y(t)=4*exp(-t)+2*exp(t),y(0) = -1, D(y)(0) = 2],y(t), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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