20.2.17 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 : Monday, January 27, 2025 at 07:47:34 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*}

✓ Solution by Maple

Time used: 0.055 (sec). Leaf size: 26

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

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

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 39

DSolve[{m*D[ v[t],t]==m*g-k*v[t]^2,{v[0]==0}},v[t],t,IncludeSingularSolutions -> True]
 

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