6.5 problem 25

Internal problem ID [6672]
Internal file name [OUTPUT/5920_Sunday_June_05_2022_04_01_38_PM_35627410/index.tex]

Book: DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section: CHAPTER 7 THE LAPLACE TRANSFORM. 7.3.1 TRANSLATION ON THE s-AXIS. Page 297
Problem number: 25.
ODE order: 2.
ODE degree: 1.

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

Maple gives the following as the ode type

[[_2nd_order, _with_linear_symmetries]]

\[ \boxed {y^{\prime \prime }-6 y^{\prime }+9 y=t} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}

6.5.1 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) &=-6\\ q(t) &=9\\ F &=t \end {align*}

Hence the ode is \begin {align*} y^{\prime \prime }-6 y^{\prime }+9 y = t \end {align*}

The domain of \(p(t)=-6\) 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 )-6 s Y \left (s \right )+6 y \left (0\right )+9 Y \left (s \right ) = \frac {1}{s^{2}}\tag {1} \end {align*}

But the initial conditions are \begin {align*} y \left (0\right )&=0\\ y'(0) &=1 \end {align*}

Substituting these initial conditions in above in Eq (1) gives \begin {align*} s^{2} Y \left (s \right )-1-6 s Y \left (s \right )+9 Y \left (s \right ) = \frac {1}{s^{2}} \end {align*}

Solving the above equation for \(Y(s)\) results in \begin {align*} Y(s) = \frac {s^{2}+1}{s^{2} \left (s^{2}-6 s +9\right )} \end {align*}

Applying partial fractions decomposition results in \[ Y(s)= -\frac {2}{27 \left (s -3\right )}+\frac {2}{27 s}+\frac {1}{9 s^{2}}+\frac {10}{9 \left (s -3\right )^{2}} \] The inverse Laplace of each term above is now found, which gives \begin {align*} \mathcal {L}^{-1}\left (-\frac {2}{27 \left (s -3\right )}\right ) &= -\frac {2 \,{\mathrm e}^{3 t}}{27}\\ \mathcal {L}^{-1}\left (\frac {2}{27 s}\right ) &= {\frac {2}{27}}\\ \mathcal {L}^{-1}\left (\frac {1}{9 s^{2}}\right ) &= \frac {t}{9}\\ \mathcal {L}^{-1}\left (\frac {10}{9 \left (s -3\right )^{2}}\right ) &= \frac {10 t \,{\mathrm e}^{3 t}}{9} \end {align*}

Adding the above results and simplifying gives \[ y=\frac {2}{27}+\frac {t}{9}+\frac {2 \,{\mathrm e}^{3 t} \left (15 t -1\right )}{27} \] Simplifying the solution gives \[ y = \frac {\left (30 t -2\right ) {\mathrm e}^{3 t}}{27}+\frac {t}{9}+\frac {2}{27} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (30 t -2\right ) {\mathrm e}^{3 t}}{27}+\frac {t}{9}+\frac {2}{27} \\ \end{align*}

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ y = \frac {\left (30 t -2\right ) {\mathrm e}^{3 t}}{27}+\frac {t}{9}+\frac {2}{27} \] Verified OK.

6.5.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [\frac {d}{d t}y^{\prime }-6 y^{\prime }+9 y=t , y \left (0\right )=0, y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=1\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 homogeneous ODE}\hspace {3pt} \\ {} & {} & r^{2}-6 r +9=0 \\ \bullet & {} & \textrm {Factor the characteristic polynomial}\hspace {3pt} \\ {} & {} & \left (r -3\right )^{2}=0 \\ \bullet & {} & \textrm {Root of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =3 \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (t \right )={\mathrm e}^{3 t} \\ \bullet & {} & \textrm {Repeated root, multiply}\hspace {3pt} y_{1}\left (t \right )\hspace {3pt}\textrm {by}\hspace {3pt} t \hspace {3pt}\textrm {to ensure linear independence}\hspace {3pt} \\ {} & {} & y_{2}\left (t \right )=t \,{\mathrm e}^{3 t} \\ \bullet & {} & \textrm {General solution of the ODE}\hspace {3pt} \\ {} & {} & y=c_{1} y_{1}\left (t \right )+c_{2} y_{2}\left (t \right )+y_{p}\left (t \right ) \\ \bullet & {} & \textrm {Substitute in solutions of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y=c_{1} {\mathrm e}^{3 t}+c_{2} t \,{\mathrm e}^{3 t}+y_{p}\left (t \right ) \\ \square & {} & \textrm {Find a particular solution}\hspace {3pt} y_{p}\left (t \right )\hspace {3pt}\textrm {of the ODE}\hspace {3pt} \\ {} & \circ & \textrm {Use variation of parameters to find}\hspace {3pt} y_{p}\hspace {3pt}\textrm {here}\hspace {3pt} f \left (t \right )\hspace {3pt}\textrm {is the forcing function}\hspace {3pt} \\ {} & {} & \left [y_{p}\left (t \right )=-y_{1}\left (t \right ) \left (\int \frac {y_{2}\left (t \right ) f \left (t \right )}{W \left (y_{1}\left (t \right ), y_{2}\left (t \right )\right )}d t \right )+y_{2}\left (t \right ) \left (\int \frac {y_{1}\left (t \right ) f \left (t \right )}{W \left (y_{1}\left (t \right ), y_{2}\left (t \right )\right )}d t \right ), f \left (t \right )=t \right ] \\ {} & \circ & \textrm {Wronskian of solutions of the homogeneous equation}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (t \right ), y_{2}\left (t \right )\right )=\left [\begin {array}{cc} {\mathrm e}^{3 t} & t \,{\mathrm e}^{3 t} \\ 3 \,{\mathrm e}^{3 t} & {\mathrm e}^{3 t}+3 t \,{\mathrm e}^{3 t} \end {array}\right ] \\ {} & \circ & \textrm {Compute Wronskian}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (t \right ), y_{2}\left (t \right )\right )={\mathrm e}^{6 t} \\ {} & \circ & \textrm {Substitute functions into equation for}\hspace {3pt} y_{p}\left (t \right ) \\ {} & {} & y_{p}\left (t \right )={\mathrm e}^{3 t} \left (-\left (\int t^{2} {\mathrm e}^{-3 t}d t \right )+\left (\int t \,{\mathrm e}^{-3 t}d t \right ) t \right ) \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y_{p}\left (t \right )=\frac {t}{9}+\frac {2}{27} \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & y=c_{1} {\mathrm e}^{3 t}+c_{2} t \,{\mathrm e}^{3 t}+\frac {t}{9}+\frac {2}{27} \\ \square & {} & \textrm {Check validity of solution}\hspace {3pt} y=c_{1} {\mathrm e}^{3 t}+c_{2} t {\mathrm e}^{3 t}+\frac {t}{9}+\frac {2}{27} \\ {} & \circ & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=0 \\ {} & {} & 0=c_{1} +\frac {2}{27} \\ {} & \circ & \textrm {Compute derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime }=3 c_{1} {\mathrm e}^{3 t}+c_{2} {\mathrm e}^{3 t}+3 c_{2} t \,{\mathrm e}^{3 t}+\frac {1}{9} \\ {} & \circ & \textrm {Use the initial condition}\hspace {3pt} y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=1 \\ {} & {} & 1=3 c_{1} +c_{2} +\frac {1}{9} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} c_{1} \hspace {3pt}\textrm {and}\hspace {3pt} c_{2} \\ {} & {} & \left \{c_{1} =-\frac {2}{27}, c_{2} =\frac {10}{9}\right \} \\ {} & \circ & \textrm {Substitute constant values into general solution and simplify}\hspace {3pt} \\ {} & {} & y=\frac {\left (30 t -2\right ) {\mathrm e}^{3 t}}{27}+\frac {t}{9}+\frac {2}{27} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\frac {\left (30 t -2\right ) {\mathrm e}^{3 t}}{27}+\frac {t}{9}+\frac {2}{27} \end {array} \]

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying high order exact linear fully integrable 
trying differential order: 2; linear nonhomogeneous with symmetry [0,1] 
trying a double symmetry of the form [xi=0, eta=F(x)] 
-> Try solving first the homogeneous part of the ODE 
   checking if the LODE has constant coefficients 
   <- constant coefficients successful 
<- solving first the homogeneous part of the ODE successful`
 

✓ Solution by Maple

Time used: 1.641 (sec). Leaf size: 20

dsolve([diff(y(t),t$2)-6*diff(y(t),t)+9*y(t)=t,y(0) = 0, D(y)(0) = 1],y(t), singsol=all)
 

\[ y \left (t \right ) = \frac {\left (30 t -2\right ) {\mathrm e}^{3 t}}{27}+\frac {t}{9}+\frac {2}{27} \]

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 25

DSolve[{y''[t]-6*y'[t]+9*y[t]==t,{y[0]==0,y'[0]==1}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{27} \left (3 t+e^{3 t} (30 t-2)+2\right ) \]