7.2 problem 1.2-2 (b)

Internal problem ID [2477]
Internal file name [OUTPUT/1969_Sunday_June_05_2022_02_41_26_AM_44077002/index.tex]

Book: Ordinary Differential Equations, Robert H. Martin, 1983
Section: Problem 1.2-2, page 12
Problem number: 1.2-2 (b).
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]

\[ \boxed {y^{\prime }-2 y=0} \] With initial conditions \begin {align*} [y \left (\ln \left (3\right )\right ) = 3] \end {align*}

7.2.1 Existence and uniqueness analysis

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

Where here \begin {align*} p(t) &=-2\\ q(t) &=0 \end {align*}

Hence the ode is \begin {align*} y^{\prime }-2 y = 0 \end {align*}

The domain of \(p(t)=-2\) is \[ \{-\infty

7.2.2 Solving as quadrature ode

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

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

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

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

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

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

Summary

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

Verification of solutions

\[ \frac {\ln \left (y\right )}{2} = t -\frac {\ln \left (3\right )}{2} \] Verified OK.

7.2.3 Maple step by step solution

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

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: 0.016 (sec). Leaf size: 10

dsolve([diff(y(t),t)=2*y(t),y(ln(3)) = 3],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.022 (sec). Leaf size: 14

DSolve[{y'[t]==2*y[t],y[Log[3]]==3},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {e^{2 t}}{3} \]