4.6 problem 1(f)

Internal problem ID [11372]
Internal file name [OUTPUT/10355_Wednesday_May_17_2023_07_49_50_PM_20057251/index.tex]

Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section: Chapter 1, First order differential equations. Section 1.3.1 Separable equations. Exercises page 26
Problem number: 1(f).
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]

\[ \boxed {Q^{\prime }-\frac {Q}{4+Q^{2}}=0} \]

4.6.1 Solving as quadrature ode

Integrating both sides gives \begin {align*} \int \frac {Q^{2}+4}{Q}d Q &= t +c_{1}\\ \frac {Q^{2}}{2}+4 \ln \left (Q \right )&=t +c_{1} \end {align*}

Solving for \(Q\) gives these solutions \begin {align*} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} Q &= {\mathrm e}^{-\frac {\operatorname {LambertW}\left (\frac {{\mathrm e}^{\frac {c_{1}}{2}+\frac {t}{2}}}{4}\right )}{2}+\frac {t}{4}+\frac {c_{1}}{4}} \\ \end{align*}

Figure 38: Slope field plot

Verification of solutions

\[ Q = {\mathrm e}^{-\frac {\operatorname {LambertW}\left (\frac {{\mathrm e}^{\frac {c_{1}}{2}+\frac {t}{2}}}{4}\right )}{2}+\frac {t}{4}+\frac {c_{1}}{4}} \] Verified OK.

4.6.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & Q^{\prime }-\frac {Q}{4+Q^{2}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & Q^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & Q^{\prime }=\frac {Q}{4+Q^{2}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {Q^{\prime } \left (4+Q^{2}\right )}{Q}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {Q^{\prime } \left (4+Q^{2}\right )}{Q}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {Q^{2}}{2}+4 \ln \left (Q\right )=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} Q \\ {} & {} & Q={\mathrm e}^{-\frac {\mathit {LambertW}\left (\frac {{\mathrm e}^{\frac {c_{1}}{2}+\frac {t}{2}}}{4}\right )}{2}+\frac {t}{4}+\frac {c_{1}}{4}} \end {array} \]

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: 38

dsolve(diff(Q(t),t)=Q(t)/(4+Q(t)^2),Q(t), singsol=all)
 

\[ Q \left (t \right ) = \frac {2 \,{\mathrm e}^{\frac {t}{4}+\frac {c_{1}}{4}}}{\sqrt {\frac {{\mathrm e}^{\frac {t}{2}+\frac {c_{1}}{2}}}{\operatorname {LambertW}\left (\frac {{\mathrm e}^{\frac {t}{2}+\frac {c_{1}}{2}}}{4}\right )}}} \]

✓ Solution by Mathematica

Time used: 0.092 (sec). Leaf size: 42

DSolve[Q'[t]==Q[t]/(4*Q[t]^2),Q[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} Q(t)\to -\frac {\sqrt {t+4 c_1}}{\sqrt {2}} \\ Q(t)\to \frac {\sqrt {t+4 c_1}}{\sqrt {2}} \\ \end{align*}