65.3.10 problem 8.6

Internal problem ID [13728]
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 : Tuesday, January 28, 2025 at 05:59:54 AM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 29

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: 0.300 (sec). Leaf size: 78

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