76.18.9 problem 20

Internal problem ID [17720]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.2 (Properties of the Laplace transform). Problems at page 309
Problem number : 20
Date solved : Tuesday, January 28, 2025 at 11:01:48 AM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime }+y^{\prime \prime }+y^{\prime }+y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=4\\ y^{\prime }\left (0\right )&=0\\ y^{\prime \prime }\left (0\right )&=-2 \end{align*}

✓ Solution by Maple

Time used: 11.694 (sec). Leaf size: 15

dsolve([diff(y(t),t$3)+diff(y(t),t$2)+diff(y(t),t)+y(t)=0,y(0) = 4, D(y)(0) = 0, (D@@2)(y)(0) = -2],y(t), singsol=all)
 

\[ y = 3 \cos \left (t \right )+\sin \left (t \right )+{\mathrm e}^{-t} \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 17

DSolve[{D[y[t],{t,3}]+D[y[t],{t,2}]+D[y[t],t]+y[t]==0,{y[0]==4,Derivative[1][y][0] == 0,Derivative[2][y][0] == -2}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{-t}+\sin (t)+3 \cos (t) \]