72.2.1 problem 1

Internal problem ID [19389]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 1. The Nature of Differential Equations. Separable Equations. Section 5. Problems at page 37
Problem number : 1
Date solved : Thursday, October 02, 2025 at 04:20:15 PM
CAS classification : [_quadrature]

\begin{align*} v^{\prime }&=g -\frac {k v^{2}}{m} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 28

ode:=diff(v(t),t) = g-k/m*v(t)^2; 
dsolve(ode,v(t), singsol=all);
 

\[ v = \frac {\tanh \left (\frac {\sqrt {g m k}\, \left (c_1 +t \right )}{m}\right ) \sqrt {g m k}}{k} \]

✓ Mathematica. Time used: 2.412 (sec). Leaf size: 85

ode=D[v[t],t]==g-k/m*v[t]^2; 
ic={}; 
DSolve[{ode,ic},v[t],t,IncludeSingularSolutions->True]
 

\begin{align*} v(t)&\to \frac {\sqrt {g} \sqrt {m} \tanh \left (\frac {\sqrt {g} \sqrt {k} (t+c_1 m)}{\sqrt {m}}\right )}{\sqrt {k}}\\ v(t)&\to -\frac {\sqrt {g} \sqrt {m}}{\sqrt {k}}\\ v(t)&\to \frac {\sqrt {g} \sqrt {m}}{\sqrt {k}} \end{align*}

✓ Sympy. Time used: 5.016 (sec). Leaf size: 42

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

\[ v{\left (t \right )} = \frac {\sqrt {g} \sqrt {m}}{\sqrt {k} \tanh {\left (\sqrt {g} \sqrt {k} \left (C_{1} \sqrt {m} + \frac {t}{\sqrt {m}}\right ) \right )}} \]