57.6.8 problem 3(d)

Internal problem ID [14400]
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 : 3(d)
Date solved : Thursday, October 02, 2025 at 09:36:42 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 )&=2 \\ \end{align*}

✓ Maple. Time used: 0.049 (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) = 2]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 21

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

\begin{align*} x(t)&\to -\frac {1}{2} e^{-3 t} \left (e^{2 t}+1\right ) \end{align*}

✓ Sympy. Time used: 0.108 (sec). Leaf size: 19

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): 2} 
dsolve(ode,func=x(t),ics=ics)
 

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