68.20.3 problem 3

Internal problem ID [17922]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 5. Applications of Higher Order Equations. Exercises 5.2, page 241
Problem number : 3
Date solved : Thursday, October 02, 2025 at 02:29:50 PM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

\begin{align*} x \left (0\right )&=-{\frac {1}{2}} \\ x^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.106 (sec). Leaf size: 35

ode:=1/4*diff(diff(x(t),t),t)+2*diff(x(t),t)+x(t) = 0; 
ic:=[x(0) = -1/2, D(x)(0) = 1]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = -\frac {{\mathrm e}^{-2 \left (2+\sqrt {3}\right ) t} \left (\left (3+\sqrt {3}\right ) {\mathrm e}^{4 t \sqrt {3}}-\sqrt {3}+3\right )}{12} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 47

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

\begin{align*} x(t)&\to \frac {1}{12} e^{-2 \left (2+\sqrt {3}\right ) t} \left (-\left (3+\sqrt {3}\right ) e^{4 \sqrt {3} t}-3+\sqrt {3}\right ) \end{align*}

✓ Sympy. Time used: 0.155 (sec). Leaf size: 48

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

\[ x{\left (t \right )} = \left (- \frac {1}{4} - \frac {\sqrt {3}}{12}\right ) e^{2 t \left (-2 + \sqrt {3}\right )} + \left (- \frac {1}{4} + \frac {\sqrt {3}}{12}\right ) e^{- 2 t \left (\sqrt {3} + 2\right )} \]