4.11.2 problem 29

Internal problem ID [1356]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 3, Second order linear equations, 3.7 Mechanical and Electrical Vibrations. page 203
Problem number : 29
Date solved : Tuesday, September 30, 2025 at 04:33:06 AM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

\begin{align*} u \left (0\right )&=0 \\ u^{\prime }\left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.098 (sec). Leaf size: 20

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

\[ u = \frac {16 \sqrt {127}\, {\mathrm e}^{-\frac {t}{8}} \sin \left (\frac {\sqrt {127}\, t}{8}\right )}{127} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 30

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

\begin{align*} u(t)&\to \frac {16 e^{-t/8} \sin \left (\frac {\sqrt {127} t}{8}\right )}{\sqrt {127}} \end{align*}

✓ Sympy. Time used: 0.132 (sec). Leaf size: 26

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

\[ u{\left (t \right )} = \frac {16 \sqrt {127} e^{- \frac {t}{8}} \sin {\left (\frac {\sqrt {127} t}{8} \right )}}{127} \]