76.19.1 problem 1

Internal problem ID [17725]
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 5.4 (Solving differential equations with Laplace transform). Problems at page 327
Problem number : 1
Date solved : Tuesday, January 28, 2025 at 11:01:53 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }-12 y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=8\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 10.803 (sec). Leaf size: 17

dsolve([diff(y(t),t$2)-4*diff(y(t),t)-12*y(t)=0,y(0) = 8, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = 2 \,{\mathrm e}^{6 t}+6 \,{\mathrm e}^{-2 t} \]

✓ Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 19

DSolve[{D[y[t],{t,2}]-4*D[y[t],t]-12*y[t]==0,{y[0]==8,Derivative[1][y][0] == 0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to 2 e^{-2 t} \left (e^{8 t}+3\right ) \]