3.10 problem 8.6

Internal problem ID [11991]
Internal file name [OUTPUT/10644_Saturday_September_02_2023_02_48_55_PM_1776475/index.tex]

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.
ODE order: 1.
ODE degree: 1.

The type(s) of ODE detected by this program : "quadrature"

Maple gives the following as the ode type

[_quadrature]

\[ \boxed {m v^{\prime }-k v^{2}=-m g} \]

3.10.1 Solving as quadrature ode

Integrating both sides gives \begin {align*} \int \frac {m}{k \,v^{2}-m g}d v &= t +c_{1}\\ -\frac {m \,\operatorname {arctanh}\left (\frac {k v}{\sqrt {m g k}}\right )}{\sqrt {m g k}}&=t +c_{1} \end {align*}

Solving for \(v\) gives these solutions \begin {align*} v_1&=-\frac {\tanh \left (\frac {\sqrt {m g k}\, \left (t +c_{1} \right )}{m}\right ) \sqrt {m g k}}{k} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} v &= -\frac {\tanh \left (\frac {\sqrt {m g k}\, \left (t +c_{1} \right )}{m}\right ) \sqrt {m g k}}{k} \\ \end{align*}

Verification of solutions

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

3.10.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & m v^{\prime }-k v^{2}=-m g \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & v^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & v^{\prime }=\frac {-m g +k v^{2}}{m} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {v^{\prime }}{-m g +k v^{2}}=\frac {1}{m} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {v^{\prime }}{-m g +k v^{2}}d t =\int \frac {1}{m}d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {\mathrm {arctanh}\left (\frac {v k}{\sqrt {m g k}}\right )}{\sqrt {m g k}}=\frac {t}{m}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} v \\ {} & {} & v=-\frac {\tanh \left (\frac {\sqrt {m g k}\, \left (c_{1} m +t \right )}{m}\right ) \sqrt {m g k}}{k} \end {array} \]

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
<- separable successful`
 

✓ Solution by Maple

Time used: 0.0 (sec). Leaf size: 29

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

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

✓ Solution by Mathematica

Time used: 14.167 (sec). Leaf size: 87

DSolve[m*v'[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*}