63.7.5 problem 1(e)

Internal problem ID [13053]
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 2.2.3 Complex eigenvalues. Exercises page 94
Problem number : 1(e)
Date solved : Monday, March 31, 2025 at 07:32:57 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 2 x^{\prime \prime }+3 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: 0.100 (sec). Leaf size: 31

ode:=2*diff(diff(x(t),t),t)+3*diff(x(t),t)+3*x(t) = 0; 
ic:=x(0) = 1, D(x)(0) = 0; 
dsolve([ode,ic],x(t), singsol=all);
 

\[ x = \frac {{\mathrm e}^{-\frac {3 t}{4}} \left (\sqrt {15}\, \sin \left (\frac {\sqrt {15}\, t}{4}\right )+5 \cos \left (\frac {\sqrt {15}\, t}{4}\right )\right )}{5} \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 42

ode=D[x[t],{t,2}]+3*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]
 

\[ x(t)\to e^{-3 t/2} \left (\sqrt {3} \sin \left (\frac {\sqrt {3} t}{2}\right )+\cos \left (\frac {\sqrt {3} t}{2}\right )\right ) \]

✓ Sympy. Time used: 0.203 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(3*x(t) + 3*Derivative(x(t), t) + 2*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)
 

\[ x{\left (t \right )} = \left (\frac {\sqrt {15} \sin {\left (\frac {\sqrt {15} t}{4} \right )}}{5} + \cos {\left (\frac {\sqrt {15} t}{4} \right )}\right ) e^{- \frac {3 t}{4}} \]