19.4 problem 28.8 (a)

Internal problem ID [13865]
Internal file name [OUTPUT/13037_Friday_February_23_2024_06_54_17_AM_26773818/index.tex]

Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section: Chapter 28. The inverse Laplace transform. Additional Exercises. page 509
Problem number: 28.8 (a).
ODE order: 2.
ODE degree: 1.

The type(s) of ODE detected by this program : "second_order_laplace", "second_order_linear_constant_coeff"

Maple gives the following as the ode type

[[_2nd_order, _missing_x]]

\[ \boxed {y^{\prime \prime }-8 y^{\prime }+17 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 12] \end {align*}

19.4.1 Existence and uniqueness analysis

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

Where here \begin {align*} p(t) &=-8\\ q(t) &=17\\ F &=0 \end {align*}

Hence the ode is \begin {align*} y^{\prime \prime }-8 y^{\prime }+17 y = 0 \end {align*}

The domain of \(p(t)=-8\) is \[ \{-\infty

Solving using the Laplace transform method. Let \begin {align*} \mathcal {L}\left (y\right ) =Y(s) \end {align*}

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 )\\ \mathcal {L}\left (y^{\prime \prime }\right ) &= s^2 Y(s) - y'(0) - s y \left (0\right ) \end {align*}

The given ode now becomes an algebraic equation in the Laplace domain \begin {align*} s^{2} Y \left (s \right )-y^{\prime }\left (0\right )-s y \left (0\right )-8 s Y \left (s \right )+8 y \left (0\right )+17 Y \left (s \right ) = 0\tag {1} \end {align*}

But the initial conditions are \begin {align*} y \left (0\right )&=3\\ y'(0) &=12 \end {align*}

Substituting these initial conditions in above in Eq (1) gives \begin {align*} s^{2} Y \left (s \right )+12-3 s -8 s Y \left (s \right )+17 Y \left (s \right ) = 0 \end {align*}

Solving the above equation for \(Y(s)\) results in \begin {align*} Y(s) = \frac {3 s -12}{s^{2}-8 s +17} \end {align*}

Applying partial fractions decomposition results in \[ Y(s)= \frac {3}{2 \left (s -4-i\right )}+\frac {3}{2 \left (s -4+i\right )} \] The inverse Laplace of each term above is now found, which gives \begin {align*} \mathcal {L}^{-1}\left (\frac {3}{2 \left (s -4-i\right )}\right ) &= \frac {3 \,{\mathrm e}^{\left (4+i\right ) t}}{2}\\ \mathcal {L}^{-1}\left (\frac {3}{2 \left (s -4+i\right )}\right ) &= \frac {3 \,{\mathrm e}^{\left (4-i\right ) t}}{2} \end {align*}

Adding the above results and simplifying gives \[ y=3 \,{\mathrm e}^{4 t} \cos \left (t \right ) \] Simplifying the solution gives \[ y = 3 \,{\mathrm e}^{4 t} \cos \left (t \right ) \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= 3 \,{\mathrm e}^{4 t} \cos \left (t \right ) \\ \end{align*}

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ y = 3 \,{\mathrm e}^{4 t} \cos \left (t \right ) \] Verified OK.

19.4.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [\frac {d}{d t}y^{\prime }-8 y^{\prime }+17 y=0, y \left (0\right )=3, y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=12\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d t}y^{\prime } \\ \bullet & {} & \textrm {Characteristic polynomial of ODE}\hspace {3pt} \\ {} & {} & r^{2}-8 r +17=0 \\ \bullet & {} & \textrm {Use quadratic formula to solve for}\hspace {3pt} r \\ {} & {} & r =\frac {8\pm \left (\sqrt {-4}\right )}{2} \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (4-\mathrm {I}, 4+\mathrm {I}\right ) \\ \bullet & {} & \textrm {1st solution of the ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (t \right )={\mathrm e}^{4 t} \cos \left (t \right ) \\ \bullet & {} & \textrm {2nd solution of the ODE}\hspace {3pt} \\ {} & {} & y_{2}\left (t \right )={\mathrm e}^{4 t} \sin \left (t \right ) \\ \bullet & {} & \textrm {General solution of the ODE}\hspace {3pt} \\ {} & {} & y=c_{1} y_{1}\left (t \right )+c_{2} y_{2}\left (t \right ) \\ \bullet & {} & \textrm {Substitute in solutions}\hspace {3pt} \\ {} & {} & y=c_{1} {\mathrm e}^{4 t} \cos \left (t \right )+c_{2} {\mathrm e}^{4 t} \sin \left (t \right ) \\ \square & {} & \textrm {Check validity of solution}\hspace {3pt} y=c_{1} {\mathrm e}^{4 t} \cos \left (t \right )+c_{2} {\mathrm e}^{4 t} \sin \left (t \right ) \\ {} & \circ & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=3 \\ {} & {} & 3=c_{1} \\ {} & \circ & \textrm {Compute derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime }=4 c_{1} {\mathrm e}^{4 t} \cos \left (t \right )-c_{1} {\mathrm e}^{4 t} \sin \left (t \right )+4 c_{2} {\mathrm e}^{4 t} \sin \left (t \right )+c_{2} {\mathrm e}^{4 t} \cos \left (t \right ) \\ {} & \circ & \textrm {Use the initial condition}\hspace {3pt} y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=12 \\ {} & {} & 12=4 c_{1} +c_{2} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} c_{1} \hspace {3pt}\textrm {and}\hspace {3pt} c_{2} \\ {} & {} & \left \{c_{1} =3, c_{2} =0\right \} \\ {} & \circ & \textrm {Substitute constant values into general solution and simplify}\hspace {3pt} \\ {} & {} & y=3 \,{\mathrm e}^{4 t} \cos \left (t \right ) \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=3 \,{\mathrm e}^{4 t} \cos \left (t \right ) \end {array} \]

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
checking if the LODE has constant coefficients 
<- constant coefficients successful`
 

✓ Solution by Maple

Time used: 4.593 (sec). Leaf size: 12

dsolve([diff(y(t),t$2)-8*diff(y(t),t)+17*y(t)=0,y(0) = 3, D(y)(0) = 12],y(t), singsol=all)
 

\[ y \left (t \right ) = 3 \,{\mathrm e}^{4 t} \cos \left (t \right ) \]

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 14

DSolve[{y''[t]-8*y'[t]+17*y[t]==0,{y[0]==3,y'[0]==12}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to 3 e^{4 t} \cos (t) \]