problem 13

Internal problem ID [838]
Internal ﬁle name [OUTPUT/838_Sunday_June_05_2022_01_50_50_AM_2433388/index.tex]

Book: Elementary diﬀerential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section: Chapter 6.2, The Laplace Transform. Solution of Initial Value Problems. page 255
Problem number: 13.
ODE order: 4.
ODE degree: 1.

\[ \boxed {y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+6 y^{\prime \prime }-4 y^{\prime }+y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1, y^{\prime \prime }\left (0\right ) = 0, y^{\prime \prime \prime }\left (0\right ) = 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 )\\ \mathcal {L}\left (y^{\prime \prime }\right ) &= s^2 Y(s) - y'(0) - s y \left (0\right )\\ \mathcal {L}\left (y^{\prime \prime \prime }\right ) &= s^3 Y(s) - y''(0) - s y'(0) - s^2 y \left (0\right )\\ \mathcal {L}\left (y^{\prime \prime \prime \prime }\right ) &= s^4 Y(s) - y'''(0) - s y''(0) - s^2 y'(0)- s^3 y \left (0\right ) \end {align*}

The given ode becomes an algebraic equation in the Laplace domain \[ s^{4} Y \left (s \right )-y^{\prime \prime \prime }\left (0\right )-s y^{\prime \prime }\left (0\right )-s^{2} y^{\prime }\left (0\right )-s^{3} y \left (0\right )-4 s^{3} Y \left (s \right )+4 y^{\prime \prime }\left (0\right )+4 s y^{\prime }\left (0\right )+4 s^{2} y \left (0\right )+6 s^{2} Y \left (s \right )-6 y^{\prime }\left (0\right )-6 s y \left (0\right )-4 s Y \left (s \right )+4 y \left (0\right )+Y \left (s \right ) = 0\tag {1} \] But the initial conditions are \begin {align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=1\\ y^{\prime \prime }\left (0\right )&=0\\ y^{\prime \prime \prime }\left (0\right )&=1 \end {align*}

Substituting these initial conditions in above in Eq (1) gives \[ s^{4} Y \left (s \right )-7-s^{2}-4 s^{3} Y \left (s \right )+4 s +6 s^{2} Y \left (s \right )-4 s Y \left (s \right )+Y \left (s \right ) = 0 \] Solving the above equation for \(Y(s)\) results in \[ Y(s) = \frac {s^{2}-4 s +7}{s^{4}-4 s^{3}+6 s^{2}-4 s +1} \] Applying partial fractions decomposition results in \[ Y(s)= \frac {1}{\left (s -1\right )^{2}}-\frac {2}{\left (s -1\right )^{3}}+\frac {4}{\left (s -1\right )^{4}} \] The inverse Laplace of each term above is now found, which gives \begin {align*} \mathcal {L}^{-1}\left (\frac {1}{\left (s -1\right )^{2}}\right ) &= t \,{\mathrm e}^{t}\\ \mathcal {L}^{-1}\left (-\frac {2}{\left (s -1\right )^{3}}\right ) &= -t^{2} {\mathrm e}^{t}\\ \mathcal {L}^{-1}\left (\frac {4}{\left (s -1\right )^{4}}\right ) &= \frac {2 t^{3} {\mathrm e}^{t}}{3} \end {align*}

Adding the above results and simplifying gives \[ y=\frac {{\mathrm e}^{t} \left (2 t^{3}-3 t^{2}+3 t \right )}{3} \]

Summary

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

Veriﬁcation of solutions

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

Maple trace

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

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

3.6 problem 13