6.1 problem 1

Internal problem ID [12670]
Internal file name [OUTPUT/11323_Friday_November_03_2023_06_30_24_AM_55103717/index.tex]

Book: Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section: Chapter 2. The Initial Value Problem. Exercises 2.3.2, page 63
Problem number: 1.
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]

\[ \boxed {-3 y+y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -1] \end {align*}

6.1.1 Existence and uniqueness analysis

This is a linear ODE. In canonical form it is written as \begin {align*} y^{\prime } + p(x)y &= q(x) \end {align*}

Where here \begin {align*} p(x) &=-3\\ q(x) &=0 \end {align*}

Hence the ode is \begin {align*} -3 y+y^{\prime } = 0 \end {align*}

The domain of \(p(x)=-3\) is \[ \{-\infty

6.1.2 Solving as quadrature ode

Integrating both sides gives \begin {align*} \int \frac {1}{3 y}d y &= \int {dx}\\ \frac {\ln \left (y \right )}{3}&= x +c_{1} \end {align*}

Initial conditions are used to solve for \(c_{1}\). Substituting \(x=0\) and \(y=-1\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} \frac {i \pi }{3} = c_{1} \end {align*}

The solutions are \begin {align*} c_{1} = \frac {i \pi }{3} \end {align*}

Trying the constant \begin {align*} c_{1} = \frac {i \pi }{3} \end {align*}

Substituting \(c_{1}\) found above in the general solution gives \begin {align*} \frac {\ln \left (y \right )}{3} = x +\frac {i \pi }{3} \end {align*}

The constant \(c_{1} = \frac {i \pi }{3}\) gives valid solution.

Summary

The solution(s) found are the following \begin{align*} \tag{1} \frac {\ln \left (y\right )}{3} &= x +\frac {i \pi }{3} \\ \end{align*}

Verification of solutions

\[ \frac {\ln \left (y\right )}{3} = x +\frac {i \pi }{3} \] Verified OK.

6.1.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [-3 y+y^{\prime }=0, y \left (0\right )=-1\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=3 y \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=3 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int 3d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=3 x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{3 x +c_{1}} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=-1 \\ {} & {} & -1={\mathrm e}^{c_{1}} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} c_{1} \\ {} & {} & c_{1} =\mathrm {I} \pi \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} c_{1} =\mathrm {I} \pi \hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & y=-{\mathrm e}^{3 x} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=-{\mathrm e}^{3 x} \end {array} \]

Maple trace

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

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 10

dsolve([diff(y(x),x)=3*y(x),y(0) = -1],y(x), singsol=all)
 

\[ y \left (x \right ) = -{\mathrm e}^{3 x} \]

✓ Solution by Mathematica

Time used: 0.039 (sec). Leaf size: 12

DSolve[{y'[x]==3*y[x],{y[0]==-1}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -e^{3 x} \]