85.36.3 problem 1 (c)

Internal problem ID [22731]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 4. Linear differential equations. A Exercises at page 171
Problem number : 1 (c)
Date solved : Thursday, October 02, 2025 at 09:14:08 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} s^{\prime \prime }+b s^{\prime }+\omega ^{2} s&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 40

ode:=diff(diff(s(t),t),t)+b*diff(s(t),t)+omega^2*s(t) = 0; 
dsolve(ode,s(t), singsol=all);
 

\[ s = \left (c_1 \,{\mathrm e}^{t \sqrt {b^{2}-4 \omega ^{2}}}+c_2 \right ) {\mathrm e}^{-\frac {\left (b +\sqrt {b^{2}-4 \omega ^{2}}\right ) t}{2}} \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 51

ode=D[s[t],{t,2}]+b*D[s[t],{t,1}]+\[Omega]^2*s[t]==0; 
ic={}; 
DSolve[{ode,ic},s[t],t,IncludeSingularSolutions->True]
 

\begin{align*} s(t)&\to e^{-\frac {1}{2} t \left (\sqrt {b^2-4 \omega ^2}+b\right )} \left (c_2 e^{t \sqrt {b^2-4 \omega ^2}}+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.171 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
b = symbols("b") 
w = symbols("w") 
s = Function("s") 
ode = Eq(b*Derivative(s(t), t) + w**2*s(t) + Derivative(s(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=s(t),ics=ics)
 

\[ s{\left (t \right )} = C_{1} e^{\frac {t \left (- b + \sqrt {b^{2} - 4 w^{2}}\right )}{2}} + C_{2} e^{- \frac {t \left (b + \sqrt {b^{2} - 4 w^{2}}\right )}{2}} \]