63.6.4 problem 1(d)

Internal problem ID [13044]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.2.2 Real eigenvalues. Exercises page 90
Problem number : 1(d)
Date solved : Monday, March 31, 2025 at 07:32:39 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+4 x^{\prime }+3 x&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.048 (sec). Leaf size: 17

ode:=diff(diff(x(t),t),t)+4*diff(x(t),t)+3*x(t) = 0; 
ic:=x(0) = 1, D(x)(0) = 0; 
dsolve([ode,ic],x(t), singsol=all);
 

\[ x = \frac {3 \,{\mathrm e}^{-t}}{2}-\frac {{\mathrm e}^{-3 t}}{2} \]

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

ode=D[x[t],{t,2}]+4*D[x[t],t]+3*x[t]==0; 
ic={x[0]==1,Derivative[1][x][0 ]==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.180 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(3*x(t) + 4*Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 1, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

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