44.4.49 problem 41

Internal problem ID [7062]
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 : Monday, January 27, 2025 at 02:46:35 PM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 28

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

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

✓ Solution by Mathematica

Time used: 3.775 (sec). Leaf size: 85

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