6.13 problem 13

Internal problem ID [5691]
Internal file name [OUTPUT/4939_Sunday_June_05_2022_03_11_05_PM_14214297/index.tex]

Book: ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section: Chapter 6. Laplace Transforms. Problem set 6.2, page 216
Problem number: 13.
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]

\[ \boxed {y^{\prime }-6 y=0} \] With initial conditions \begin {align*} [y \left (-1\right ) = 4] \end {align*}

6.13.1 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) &=-6\\ q(t) &=0 \end {align*}

Hence the ode is \begin {align*} y^{\prime }-6 y = 0 \end {align*}

The domain of \(p(t)=-6\) is \[ \{-\infty

6.13.2 Solving as laplace ode

Since initial condition is not at zero, then let \begin {align*} y \left (0\right ) = c_{1} \end {align*}

Solving using the Laplace transform method. Let \[ \mathcal {L}\left (y\right ) =Y(s) \] Taking the Laplace transform of the ode and using the relations that \begin {align*} \mathcal {L}\left (y^{\prime }\right )&= s Y(s) - y \left (0\right ) \end {align*}

The given ode now becomes an algebraic equation in the Laplace domain \begin {align*} s Y \left (s \right )-y \left (0\right )-6 Y \left (s \right ) = 0\tag {1} \end {align*}

Replacing initial condition gives \begin {align*} s Y \left (s \right )-c_{1} -6 Y \left (s \right ) = 0 \end {align*}

Solving for \(Y(s)\) gives \begin {align*} Y(s) = \frac {c_{1}}{s -6} \end {align*}

Taking the inverse Laplace transform gives \begin {align*} y&= \mathcal {L}^{-1}\left (Y(s)\right )\\ &= \mathcal {L}^{-1}\left (\frac {c_{1}}{s -6}\right )\\ &= c_{1} {\mathrm e}^{6 t} \end {align*}

The constant \(c_1\) is determined from the given initial condition \(y(0)=c_{1}\) using the solution found above. This results in \begin {align*} 4&=c_{1} {\mathrm e}^{-6} \end {align*}

Solving gives \begin {align*} c_1 &=4 \,{\mathrm e}^{6} \end {align*}

Hence the solution now becomes \begin {align*} y&=4 \,{\mathrm e}^{6} {\mathrm e}^{6 t} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= 4 \,{\mathrm e}^{6} {\mathrm e}^{6 t} \\ \end{align*}

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ y = 4 \,{\mathrm e}^{6} {\mathrm e}^{6 t} \] Verified OK.

6.13.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime }-6 y=0, y \left (-1\right )=4\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=6 y \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=6 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{y}d t =\int 6d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=6 t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{6 t +c_{1}} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (-1\right )=4 \\ {} & {} & 4={\mathrm e}^{-6+c_{1}} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} c_{1} \\ {} & {} & c_{1} =6+2 \ln \left (2\right ) \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} c_{1} =6+2 \ln \left (2\right )\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & y=4 \,{\mathrm e}^{6 t +6} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=4 \,{\mathrm e}^{6 t +6} \end {array} \]

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: 0.625 (sec). Leaf size: 12

dsolve([diff(y(t),t)-6*y(t)=0,y(-1) = 4],y(t), singsol=all)
 

\[ y \left (t \right ) = 4 \,{\mathrm e}^{6 t +6} \]

✓ Solution by Mathematica

Time used: 0.023 (sec). Leaf size: 14

DSolve[{y'[t]-6*y[t]==0,{y[-1]==4}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to 4 e^{6 t+6} \]