14.2.17 problem Problem 17

Internal problem ID [3609]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 1, First-Order Differential Equations. Section 1.4, Separable Differential Equations. page 43
Problem number : Problem 17
Date solved : Tuesday, September 30, 2025 at 06:48:28 AM
CAS classification : [_quadrature]

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

With initial conditions

\begin{align*} v \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.071 (sec). Leaf size: 26

ode:=m*diff(v(t),t) = m*g-k*v(t)^2; 
ic:=[v(0) = 0]; 
dsolve([ode,op(ic)],v(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 39

ode=m*D[ v[t],t]==m*g-k*v[t]^2; 
ic={v[0]==0}; 
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}{\sqrt {m}}\right )}{\sqrt {k}} \end{align*}

✗ Sympy

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 = {v(0): 0} 
dsolve(ode,func=v(t),ics=ics)
 

NotImplementedError : Initial conditions produced too many solutions for constants