problem 31

Internal problem ID [13226]
Internal ﬁle name [OUTPUT/11882_Tuesday_December_05_2023_12_12_42_PM_12105076/index.tex]

Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 6. Laplace transform. Section 6.3 page 600
Problem number: 31.
ODE order: 2.
ODE degree: 1.

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

\[ \boxed {y^{\prime \prime }+4 y=\cos \left (2 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = -2, y^{\prime }\left (0\right ) = 0] \end {align*}

19.5.1 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) &=0\\ q(t) &=4\\ F &=\cos \left (2 t \right ) \end {align*}

Hence the ode is \begin {align*} y^{\prime \prime }+4 y = \cos \left (2 t \right ) \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 )+4 Y \left (s \right ) = \frac {s}{s^{2}+4}\tag {1} \end {align*}

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

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

Solving the above equation for \(Y(s)\) results in \begin {align*} Y(s) = -\frac {s \left (2 s^{2}+7\right )}{\left (s^{2}+4\right )^{2}} \end {align*}

Applying partial fractions decomposition results in \[ Y(s)= -\frac {i}{8 \left (s -2 i\right )^{2}}+\frac {i}{8 \left (s +2 i\right )^{2}}-\frac {1}{s -2 i}-\frac {1}{s +2 i} \] The inverse Laplace of each term above is now found, which gives \begin {align*} \mathcal {L}^{-1}\left (-\frac {i}{8 \left (s -2 i\right )^{2}}\right ) &= -\frac {i t \,{\mathrm e}^{2 i t}}{8}\\ \mathcal {L}^{-1}\left (\frac {i}{8 \left (s +2 i\right )^{2}}\right ) &= \frac {i t \,{\mathrm e}^{-2 i t}}{8}\\ \mathcal {L}^{-1}\left (-\frac {1}{s -2 i}\right ) &= -{\mathrm e}^{2 i t}\\ \mathcal {L}^{-1}\left (-\frac {1}{s +2 i}\right ) &= -{\mathrm e}^{-2 i t} \end {align*}

Adding the above results and simplifying gives \[ y=-2 \cos \left (2 t \right )+\frac {t \sin \left (2 t \right )}{4} \] Simplifying the solution gives \[ y = -2 \cos \left (2 t \right )+\frac {t \sin \left (2 t \right )}{4} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= -2 \cos \left (2 t \right )+\frac {t \sin \left (2 t \right )}{4} \\ \end{align*}

Veriﬁcation of solutions

\[ y = -2 \cos \left (2 t \right )+\frac {t \sin \left (2 t \right )}{4} \] Veriﬁed OK.

19.5.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [\frac {d}{d t}y^{\prime }+4 y=\cos \left (2 t \right ), y \left (0\right )=-2, y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=0\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}+4=0 \\ \bullet & {} & \textrm {Use quadratic formula to solve for}\hspace {3pt} r \\ {} & {} & r =\frac {0\pm \left (\sqrt {-16}\right )}{2} \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (-2 \,\mathrm {I}, 2 \,\mathrm {I}\right ) \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (t \right )=\cos \left (2 t \right ) \\ \bullet & {} & \textrm {2nd solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{2}\left (t \right )=\sin \left (2 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 )+y_{p}\left (t \right ) \\ \bullet & {} & \textrm {Substitute in solutions of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y=c_{1} \cos \left (2 t \right )+c_{2} \sin \left (2 t \right )+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 )=\cos \left (2 t \right )\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} \cos \left (2 t \right ) & \sin \left (2 t \right ) \\ -2 \sin \left (2 t \right ) & 2 \cos \left (2 t \right ) \end {array}\right ] \\ {} & \circ & \textrm {Compute Wronskian}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (t \right ), y_{2}\left (t \right )\right )=2 \\ {} & \circ & \textrm {Substitute functions into equation for}\hspace {3pt} y_{p}\left (t \right ) \\ {} & {} & y_{p}\left (t \right )=-\frac {\cos \left (2 t \right ) \left (\int \sin \left (4 t \right )d t \right )}{4}+\frac {\sin \left (2 t \right ) \left (\int \cos \left (2 t \right )^{2}d t \right )}{2} \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y_{p}\left (t \right )=\frac {\cos \left (2 t \right )}{16}+\frac {t \sin \left (2 t \right )}{4} \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & y=c_{1} \cos \left (2 t \right )+c_{2} \sin \left (2 t \right )+\frac {\cos \left (2 t \right )}{16}+\frac {t \sin \left (2 t \right )}{4} \\ \square & {} & \textrm {Check validity of solution}\hspace {3pt} y=c_{1} \cos \left (2 t \right )+c_{2} \sin \left (2 t \right )+\frac {\cos \left (2 t \right )}{16}+\frac {t \sin \left (2 t \right )}{4} \\ {} & \circ & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=-2 \\ {} & {} & -2=c_{1} +\frac {1}{16} \\ {} & \circ & \textrm {Compute derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime }=-2 c_{1} \sin \left (2 t \right )+2 c_{2} \cos \left (2 t \right )+\frac {\sin \left (2 t \right )}{8}+\frac {t \cos \left (2 t \right )}{2} \\ {} & \circ & \textrm {Use the initial condition}\hspace {3pt} y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=0 \\ {} & {} & 0=2 c_{2} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} c_{1} \hspace {3pt}\textrm {and}\hspace {3pt} c_{2} \\ {} & {} & \left \{c_{1} =-\frac {33}{16}, c_{2} =0\right \} \\ {} & \circ & \textrm {Substitute constant values into general solution and simplify}\hspace {3pt} \\ {} & {} & y=-2 \cos \left (2 t \right )+\frac {t \sin \left (2 t \right )}{4} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=-2 \cos \left (2 t \right )+\frac {t \sin \left (2 t \right )}{4} \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`

\[ y \left (t \right ) = -2 \cos \left (2 t \right )+\frac {t \sin \left (2 t \right )}{4} \]

19.5 problem 31

19.5.1 Existence and uniqueness analysis

19.5.2 Maple step by step solution