78.2.1 problem 1

Internal problem ID [18103]
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 : Tuesday, January 28, 2025 at 11:26:23 AM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 28

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

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

✓ Solution by Mathematica

Time used: 3.862 (sec). Leaf size: 85

DSolve[D[v[t],t]==g-k/m*v[t]^2,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*}