89.13.21 problem 21

Internal problem ID [24604]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 8. Linear Differential Equations with constant coefficients. Exercises at page 127
Problem number : 21
Date solved : Thursday, October 02, 2025 at 10:46:27 PM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

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

✓ Maple. Time used: 0.040 (sec). Leaf size: 36

ode:=diff(diff(x(t),t),t)+2*b*diff(x(t),t)+k^2*x(t) = 0; 
ic:=[x(0) = 0, D(x)(0) = v__0]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {v_{0} {\mathrm e}^{-t b} \sinh \left (t \sqrt {b^{2}-k^{2}}\right )}{\sqrt {b^{2}-k^{2}}} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 63

ode=D[x[t],{t,2}]+2*b*D[x[t],{t,1}]+k^2*x[t] ==0; 
ic={x[0]==0,Derivative[1][x][0] ==v0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {\text {v0} e^{-\left (t \left (\sqrt {b^2-k^2}+b\right )\right )} \left (e^{2 t \sqrt {b^2-k^2}}-1\right )}{2 \sqrt {b^2-k^2}} \end{align*}

✓ Sympy. Time used: 0.148 (sec). Leaf size: 60

from sympy import * 
t = symbols("t") 
v0 = symbols("v0") 
x = Function("x") 
ode = Eq(2*b*Derivative(x(t), t) + k**2*x(t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): v0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {v_{0} e^{t \left (- b + \sqrt {b^{2} - k^{2}}\right )}}{2 \sqrt {b^{2} - k^{2}}} - \frac {v_{0} e^{- t \left (b + \sqrt {b^{2} - k^{2}}\right )}}{2 \sqrt {b^{2} - k^{2}}} \]