65.3.10 problem 8.6

Internal problem ID [13657]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 8, Separable equations. Exercises page 72
Problem number : 8.6
Date solved : Monday, March 31, 2025 at 08:04:16 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 29

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: 0.3 (sec). Leaf size: 78

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: 7.723 (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)
 

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