86.6.6 problem 6

Internal problem ID [23139]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 5. Linear equations of the second order with constant coefficients. Exercise 5b at page 77
Problem number : 6
Date solved : Thursday, October 02, 2025 at 09:23:25 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+2 x^{\prime }+4 x&=0 \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=5 \\ x \left (\frac {\pi \sqrt {3}}{6}\right )&=2 \,{\mathrm e}^{-\frac {\pi \sqrt {3}}{6}} \\ \end{align*}

✓ Maple. Time used: 0.041 (sec). Leaf size: 26

ode:=diff(diff(x(t),t),t)+2*diff(x(t),t)+4*x(t) = 0; 
ic:=[x(0) = 5, x(1/6*3^(1/2)*Pi) = 2*exp(-1/6*3^(1/2)*Pi)]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = {\mathrm e}^{-t} \left (2 \sin \left (\sqrt {3}\, t \right )+5 \cos \left (\sqrt {3}\, t \right )\right ) \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 32

ode=D[x[t],{t,2}]+2*D[x[t],t]+4*x[t]==0; 
ic={x[0]==5,x[Pi/(2*Sqrt[3])]== 2*Exp[-Pi/(2*Sqrt[3])] }; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{-t} \left (2 \sin \left (\sqrt {3} t\right )+5 \cos \left (\sqrt {3} t\right )\right ) \end{align*}

✓ Sympy. Time used: 0.124 (sec). Leaf size: 63

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(4*x(t) + 2*Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 5, x(sqrt(3)*Pi/6): 2*exp(-sqrt(3)*pi/6)} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (\left (\frac {2 e^{\frac {\sqrt {3} \Pi }{6}}}{e^{\frac {\sqrt {3} \pi }{6}} \sin {\left (\frac {\Pi }{2} \right )}} - \frac {5 \cos {\left (\frac {\Pi }{2} \right )}}{\sin {\left (\frac {\Pi }{2} \right )}}\right ) \sin {\left (\sqrt {3} t \right )} + 5 \cos {\left (\sqrt {3} t \right )}\right ) e^{- t} \]