4.13 problem 4(e)

Internal problem ID [11379]
Internal file name [OUTPUT/10362_Wednesday_May_17_2023_07_49_59_PM_52105283/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: 4(e).
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]

\[ \boxed {y^{\prime }+y+\frac {1}{y}=0} \]

4.13.1 Solving as quadrature ode

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

Solving for \(y\) gives these solutions \begin {align*} y_1&=\sqrt {-1+{\mathrm e}^{-2 t -2 c_{1}}}\\ &=\sqrt {-1+\frac {{\mathrm e}^{-2 t}}{c_{1}^{2}}}\\ y_2&=-\sqrt {-1+{\mathrm e}^{-2 t -2 c_{1}}}\\ &=-\sqrt {-1+\frac {{\mathrm e}^{-2 t}}{c_{1}^{2}}} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \sqrt {-1+\frac {{\mathrm e}^{-2 t}}{c_{1}^{2}}} \\ \tag{2} y &= -\sqrt {-1+\frac {{\mathrm e}^{-2 t}}{c_{1}^{2}}} \\ \end{align*}

Figure 58: Slope field plot

Verification of solutions

\[ y = \sqrt {-1+\frac {{\mathrm e}^{-2 t}}{c_{1}^{2}}} \] Verified OK.

\[ y = -\sqrt {-1+\frac {{\mathrm e}^{-2 t}}{c_{1}^{2}}} \] Verified OK.

4.13.2 Maple step by step solution

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

Maple trace

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

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 29

dsolve(diff(y(t),t)+y(t)+1/y(t)=0,y(t), singsol=all)
 

\begin{align*} y \left (t \right ) &= \sqrt {{\mathrm e}^{-2 t} c_{1} -1} \\ y \left (t \right ) &= -\sqrt {{\mathrm e}^{-2 t} c_{1} -1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 4.571 (sec). Leaf size: 57

DSolve[y'[t]+y[t]+1/y[t]==0,y[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} y(t)\to -\sqrt {-1+e^{-2 t+2 c_1}} \\ y(t)\to \sqrt {-1+e^{-2 t+2 c_1}} \\ y(t)\to -i \\ y(t)\to i \\ \end{align*}