30.11.2 problem 2

Internal problem ID [7582]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 4, Linear Second-Order Equations. EXERCISES 4.1 at page 156
Problem number : 2
Date solved : Tuesday, September 30, 2025 at 04:54:35 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} m y^{\prime \prime }+b y^{\prime }+k y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 44

ode:=m*diff(diff(y(t),t),t)+b*diff(y(t),t)+k*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 55

ode=m*D[y[t],{t,2}]+b*D[y[t],t]+k*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.173 (sec). Leaf size: 46

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

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