7.18.4 problem 4

Internal problem ID [533]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 4. Laplace transform methods. Section 4.2 (Transformation of initial value problems). Problems at page 287
Problem number : 4
Date solved : Tuesday, March 04, 2025 at 11:26:16 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+8 x^{\prime }+15 x&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} x \left (0\right )&=2\\ x^{\prime }\left (0\right )&=-3 \end{align*}

✓ Maple. Time used: 0.200 (sec). Leaf size: 18

ode:=diff(diff(x(t),t),t)+8*diff(x(t),t)+15*x(t) = 0; 
ic:=x(0) = 2, D(x)(0) = -3; 
dsolve([ode,ic],x(t),method='laplace');
 

\[ x \left (t \right ) = {\mathrm e}^{-4 t} \left (2 \cosh \left (t \right )+5 \sinh \left (t \right )\right ) \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 23

ode=D[x[t],{t,2}]+8*D[x[t],t]+15*x[t]==0; 
ic={x[0]==2,Derivative[1][x][0] ==-3}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to \frac {1}{2} e^{-5 t} \left (7 e^{2 t}-3\right ) \]

✓ Sympy. Time used: 0.227 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(15*x(t) + 8*Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 2, Subs(Derivative(x(t), t), t, 0): -3} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (\frac {7}{2} - \frac {3 e^{- 2 t}}{2}\right ) e^{- 3 t} \]