problem 11

Internal problem ID [5689]
Internal ﬁle name [OUTPUT/4937_Sunday_June_05_2022_03_11_03_PM_24778006/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: 11.
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"

\[ \boxed {y^{\prime \prime }+3 y^{\prime }+\frac {9 y}{4}=9 t^{3}+64} \] With initial conditions \begin {align*} \left [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = {\frac {63}{2}}\right ] \end {align*}

6.11.1 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) &=3\\ q(t) &={\frac {9}{4}}\\ F &=9 t^{3}+64 \end {align*}

Hence the ode is \begin {align*} y^{\prime \prime }+3 y^{\prime }+\frac {9 y}{4} = 9 t^{3}+64 \end {align*}

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 )+3 s Y \left (s \right )-3 y \left (0\right )+\frac {9 Y \left (s \right )}{4} = \frac {54}{s^{4}}+\frac {64}{s}\tag {1} \end {align*}

But the initial conditions are \begin {align*} y \left (0\right )&=1\\ y'(0) &={\frac {63}{2}} \end {align*}

Substituting these initial conditions in above in Eq (1) gives \begin {align*} s^{2} Y \left (s \right )-\frac {69}{2}-s +3 s Y \left (s \right )+\frac {9 Y \left (s \right )}{4} = \frac {54}{s^{4}}+\frac {64}{s} \end {align*}

Solving the above equation for \(Y(s)\) results in \begin {align*} Y(s) = \frac {4 s^{5}+138 s^{4}+256 s^{3}+216}{s^{4} \left (4 s^{2}+12 s +9\right )} \end {align*}

Applying partial fractions decomposition results in \[ Y(s)= \frac {1}{\left (s +\frac {3}{2}\right )^{2}}+\frac {1}{s +\frac {3}{2}}-\frac {32}{s^{3}}+\frac {24}{s^{4}}+\frac {32}{s^{2}} \] The inverse Laplace of each term above is now found, which gives \begin {align*} \mathcal {L}^{-1}\left (\frac {1}{\left (s +\frac {3}{2}\right )^{2}}\right ) &= t \,{\mathrm e}^{-\frac {3 t}{2}}\\ \mathcal {L}^{-1}\left (\frac {1}{s +\frac {3}{2}}\right ) &= {\mathrm e}^{-\frac {3 t}{2}}\\ \mathcal {L}^{-1}\left (-\frac {32}{s^{3}}\right ) &= -16 t^{2}\\ \mathcal {L}^{-1}\left (\frac {24}{s^{4}}\right ) &= 4 t^{3}\\ \mathcal {L}^{-1}\left (\frac {32}{s^{2}}\right ) &= 32 t \end {align*}

Adding the above results and simplifying gives \[ y=4 t^{3}-16 t^{2}+32 t +{\mathrm e}^{-\frac {3 t}{2}} \left (t +1\right ) \] Simplifying the solution gives \[ y = 4 t^{3}+t \,{\mathrm e}^{-\frac {3 t}{2}}-16 t^{2}+{\mathrm e}^{-\frac {3 t}{2}}+32 t \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= 4 t^{3}+t \,{\mathrm e}^{-\frac {3 t}{2}}-16 t^{2}+{\mathrm e}^{-\frac {3 t}{2}}+32 t \\ \end{align*}

Veriﬁcation of solutions

\[ y = 4 t^{3}+t \,{\mathrm e}^{-\frac {3 t}{2}}-16 t^{2}+{\mathrm e}^{-\frac {3 t}{2}}+32 t \] Veriﬁed OK.

6.11.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime \prime }+3 y^{\prime }+\frac {9 y}{4}=9 t^{3}+64, y \left (0\right )=1, y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=\frac {63}{2}\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & y^{\prime \prime } \\ \bullet & {} & \textrm {Characteristic polynomial of homogeneous ODE}\hspace {3pt} \\ {} & {} & r^{2}+3 r +\frac {9}{4}=0 \\ \bullet & {} & \textrm {Factor the characteristic polynomial}\hspace {3pt} \\ {} & {} & \frac {\left (2 r +3\right )^{2}}{4}=0 \\ \bullet & {} & \textrm {Root of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =-\frac {3}{2} \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (t \right )={\mathrm e}^{-\frac {3 t}{2}} \\ \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}^{-\frac {3 t}{2}} \\ \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}^{-\frac {3 t}{2}}+c_{2} t \,{\mathrm e}^{-\frac {3 t}{2}}+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 )=9 t^{3}+64\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}^{-\frac {3 t}{2}} & t \,{\mathrm e}^{-\frac {3 t}{2}} \\ -\frac {3 \,{\mathrm e}^{-\frac {3 t}{2}}}{2} & {\mathrm e}^{-\frac {3 t}{2}}-\frac {3 t \,{\mathrm e}^{-\frac {3 t}{2}}}{2} \end {array}\right ] \\ {} & \circ & \textrm {Compute Wronskian}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (t \right ), y_{2}\left (t \right )\right )={\mathrm e}^{-3 t} \\ {} & \circ & \textrm {Substitute functions into equation for}\hspace {3pt} y_{p}\left (t \right ) \\ {} & {} & y_{p}\left (t \right )={\mathrm e}^{-\frac {3 t}{2}} \left (-\left (\int \left (9 t^{4}+64 t \right ) {\mathrm e}^{\frac {3 t}{2}}d t \right )+\left (\int {\mathrm e}^{\frac {3 t}{2}} \left (9 t^{3}+64\right )d t \right ) t \right ) \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y_{p}\left (t \right )=4 t \left (t^{2}-4 t +8\right ) \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & y=c_{1} {\mathrm e}^{-\frac {3 t}{2}}+c_{2} t \,{\mathrm e}^{-\frac {3 t}{2}}+4 t \left (t^{2}-4 t +8\right ) \\ \square & {} & \textrm {Check validity of solution}\hspace {3pt} y=c_{1} {\mathrm e}^{-\frac {3 t}{2}}+c_{2} t {\mathrm e}^{-\frac {3 t}{2}}+4 t \left (t^{2}-4 t +8\right ) \\ {} & \circ & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=1 \\ {} & {} & 1=c_{1} \\ {} & \circ & \textrm {Compute derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {3 c_{1} {\mathrm e}^{-\frac {3 t}{2}}}{2}+c_{2} {\mathrm e}^{-\frac {3 t}{2}}-\frac {3 c_{2} t \,{\mathrm e}^{-\frac {3 t}{2}}}{2}+4 t^{2}-16 t +32+4 t \left (2 t -4\right ) \\ {} & \circ & \textrm {Use the initial condition}\hspace {3pt} y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=\frac {63}{2} \\ {} & {} & \frac {63}{2}=-\frac {3 c_{1}}{2}+c_{2} +32 \\ {} & \circ & \textrm {Solve for}\hspace {3pt} c_{1} \hspace {3pt}\textrm {and}\hspace {3pt} c_{2} \\ {} & {} & \left \{c_{1} =1, c_{2} =1\right \} \\ {} & \circ & \textrm {Substitute constant values into general solution and simplify}\hspace {3pt} \\ {} & {} & y=4 t^{3}+t \,{\mathrm e}^{-\frac {3 t}{2}}-16 t^{2}+{\mathrm e}^{-\frac {3 t}{2}}+32 t \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=4 t^{3}+t \,{\mathrm e}^{-\frac {3 t}{2}}-16 t^{2}+{\mathrm e}^{-\frac {3 t}{2}}+32 t \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`

\[ y \left (t \right ) = 4 t^{3}+{\mathrm e}^{-\frac {3 t}{2}} t -16 t^{2}+{\mathrm e}^{-\frac {3 t}{2}}+32 t \]

6.11 problem 11

6.11.1 Existence and uniqueness analysis

6.11.2 Maple step by step solution