14.16.3 problem 17

Internal problem ID [2673]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.9, The method of Laplace transform. Excercises page 232
Problem number : 17
Date solved : Monday, January 27, 2025 at 06:06:14 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+y&={\mathrm e}^{-t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=3 \end{align*}

✓ Solution by Maple

Time used: 0.822 (sec). Leaf size: 18

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

\[ y = \frac {{\mathrm e}^{-t} \left (t^{2}+8 t +2\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.027 (sec). Leaf size: 22

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

\[ y(t)\to \frac {1}{2} e^{-t} \left (t^2+8 t+2\right ) \]