14.17.5 problem 23

Internal problem ID [2683]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.10, Some useful properties of Laplace transform. Excercises page 238
Problem number : 23
Date solved : Monday, January 27, 2025 at 06:06:21 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+y^{\prime }+y&=1+{\mathrm e}^{-t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=-5 \end{align*}

✓ Solution by Maple

Time used: 0.973 (sec). Leaf size: 38

dsolve([diff(y(t),t$2)+diff(y(t),t)+y(t)=1+exp(-t),y(0) = 3, D(y)(0) = -5],y(t), singsol=all)
 

\[ y = -\frac {7 \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}}{3}+{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right )+{\mathrm e}^{-t}+1 \]

✓ Solution by Mathematica

Time used: 0.760 (sec). Leaf size: 56

DSolve[{D[y[t],{t,2}]+D[y[t],t]+y[t]==1+Exp[-t],{y[0]==3,Derivative[1][y][0] ==-5}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{-t}-\frac {7 e^{-t/2} \sin \left (\frac {\sqrt {3} t}{2}\right )}{\sqrt {3}}+e^{-t/2} \cos \left (\frac {\sqrt {3} t}{2}\right )+1 \]