38.4.49 problem 41

Internal problem ID [8348]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.1 Solution curves without a solution. Exercises 2.1 at page 44
Problem number : 41
Date solved : Tuesday, September 30, 2025 at 05:29:30 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 28

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

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

✓ Mathematica. Time used: 0.173 (sec). Leaf size: 77

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

\begin{align*} v(t)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{g m-k K[1]^2}dK[1]\&\right ]\left [\frac {t}{m}+c_1\right ]\\ 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: 5.037 (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*m + k*v(t)**2 + m*Derivative(v(t), t),0) 
ics = {} 
dsolve(ode,func=v(t),ics=ics)
 

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