3.13 problem 13 (a)

Internal problem ID [14136]
Internal file name [OUTPUT/13817_Saturday_March_02_2024_02_50_56_PM_52851333/index.tex]

Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section: Chapter 2. First Order Equations. Exercises 2.1, page 32
Problem number: 13 (a).
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]

\[ \boxed {y^{\prime }-\sqrt {y^{2}-1}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2] \end {align*}

3.13.1 Existence and uniqueness analysis

This is non linear first order ODE. In canonical form it is written as \begin {align*} y^{\prime } &= f(t,y)\\ &= \sqrt {y^{2}-1} \end {align*}

The \(y\) domain of \(f(t,y)\) when \(t=0\) is \[ \{1\le y \le \infty , -\infty \le y \le -1\} \] And the point \(y_0 = 2\) is inside this domain. Now we will look at the continuity of \begin {align*} \frac {\partial f}{\partial y} &= \frac {\partial }{\partial y}\left (\sqrt {y^{2}-1}\right ) \\ &= \frac {y}{\sqrt {y^{2}-1}} \end {align*}

The \(y\) domain of \(\frac {\partial f}{\partial y}\) when \(t=0\) is \[ \{-\infty \le y <-1, -1

3.13.2 Solving as quadrature ode

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

Initial conditions are used to solve for \(c_{1}\). Substituting \(t=0\) and \(y=2\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} \ln \left (2+\sqrt {3}\right ) = c_{1} \end {align*}

The solutions are \begin {align*} c_{1} = \ln \left (2+\sqrt {3}\right ) \end {align*}

Trying the constant \begin {align*} c_{1} = \ln \left (2+\sqrt {3}\right ) \end {align*}

Substituting \(c_{1}\) found above in the general solution gives \begin {align*} \ln \left (y +\sqrt {y^{2}-1}\right ) = t +\ln \left (2+\sqrt {3}\right ) \end {align*}

The constant \(c_{1} = \ln \left (2+\sqrt {3}\right )\) gives valid solution.

Summary

The solution(s) found are the following \begin{align*} \tag{1} \ln \left (y+\sqrt {y^{2}-1}\right ) &= t +\ln \left (2+\sqrt {3}\right ) \\ \end{align*}

Verification of solutions

\[ \ln \left (y+\sqrt {y^{2}-1}\right ) = t +\ln \left (2+\sqrt {3}\right ) \] Verified OK.

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: 0.734 (sec). Leaf size: 27

dsolve([diff(y(t),t)=sqrt(y(t)^2-1),y(0) = 2],y(t), singsol=all)
 

\[ y \left (t \right ) = \frac {{\mathrm e}^{t} \sqrt {3}}{2}+{\mathrm e}^{t}-\frac {\sqrt {3}\, {\mathrm e}^{-t}}{2}+{\mathrm e}^{-t} \]

✓ Solution by Mathematica

Time used: 0.067 (sec). Leaf size: 49

DSolve[{y'[t]==Sqrt[y[t]^2-1],{y[0]==2}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{2} e^{-t} \sqrt {2 e^{2 t}+\left (7+4 \sqrt {3}\right ) e^{4 t}+7-4 \sqrt {3}} \]