4.10 problem 11(c) k=1/2

Internal problem ID [854]
Internal file name [OUTPUT/854_Sunday_June_05_2022_01_51_52_AM_90207999/index.tex]

Book: Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section: Chapter 6.4, The Laplace Transform. Differential equations with discontinuous forcing functions. page 268
Problem number: 11(c) k=1/2.
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, _linear, _nonhomogeneous]]

\[ \boxed {u^{\prime \prime }+\frac {u^{\prime }}{4}+u=\frac {\operatorname {Heaviside}\left (t -\frac {3}{2}\right )}{2}-\frac {\operatorname {Heaviside}\left (t -\frac {5}{2}\right )}{2}} \] With initial conditions \begin {align*} [u \left (0\right ) = 0, u^{\prime }\left (0\right ) = 0] \end {align*}

4.10.1 Existence and uniqueness analysis

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

Where here \begin {align*} p(t) &={\frac {1}{4}}\\ q(t) &=1\\ F &=\frac {\operatorname {Heaviside}\left (t -\frac {3}{2}\right )}{2}-\frac {\operatorname {Heaviside}\left (t -\frac {5}{2}\right )}{2} \end {align*}

Hence the ode is \begin {align*} u^{\prime \prime }+\frac {u^{\prime }}{4}+u = \frac {\operatorname {Heaviside}\left (t -\frac {3}{2}\right )}{2}-\frac {\operatorname {Heaviside}\left (t -\frac {5}{2}\right )}{2} \end {align*}

The domain of \(p(t)={\frac {1}{4}}\) is \[ \{-\infty

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

Taking the Laplace transform of the ode and using the relations that \begin {align*} \mathcal {L}\left (u^{\prime }\right ) &= s Y(s) - u \left (0\right )\\ \mathcal {L}\left (u^{\prime \prime }\right ) &= s^2 Y(s) - u'(0) - s u \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 )-u^{\prime }\left (0\right )-s u \left (0\right )+\frac {s Y \left (s \right )}{4}-\frac {u \left (0\right )}{4}+Y \left (s \right ) = \frac {{\mathrm e}^{-\frac {3 s}{2}}-{\mathrm e}^{-\frac {5 s}{2}}}{2 s}\tag {1} \end {align*}

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

Substituting these initial conditions in above in Eq (1) gives \begin {align*} s^{2} Y \left (s \right )+\frac {s Y \left (s \right )}{4}+Y \left (s \right ) = \frac {{\mathrm e}^{-\frac {3 s}{2}}-{\mathrm e}^{-\frac {5 s}{2}}}{2 s} \end {align*}

Solving the above equation for \(Y(s)\) results in \begin {align*} Y(s) = \frac {2 \,{\mathrm e}^{-\frac {3 s}{2}}-2 \,{\mathrm e}^{-\frac {5 s}{2}}}{s \left (4 s^{2}+s +4\right )} \end {align*}

Taking the inverse Laplace transform gives \begin {align*} u&= \mathcal {L}^{-1}\left (Y(s)\right )\\ &= \mathcal {L}^{-1}\left (\frac {2 \,{\mathrm e}^{-\frac {3 s}{2}}-2 \,{\mathrm e}^{-\frac {5 s}{2}}}{s \left (4 s^{2}+s +4\right )}\right )\\ &= \frac {\left (i \sqrt {7}+21\right ) \left (\left (-63+3 i \sqrt {7}+32 \,{\mathrm e}^{-\frac {\left (3 i \sqrt {7}+1\right ) \left (2 t -5\right )}{16}}+\left (31-3 i \sqrt {7}\right ) {\mathrm e}^{-\frac {\left (-3 i \sqrt {7}+1\right ) \left (2 t -5\right )}{16}}\right ) \operatorname {Heaviside}\left (t -\frac {5}{2}\right )+\left (63-3 i \sqrt {7}-32 \,{\mathrm e}^{-\frac {\left (3 i \sqrt {7}+1\right ) \left (2 t -3\right )}{16}}-\left (31-3 i \sqrt {7}\right ) {\mathrm e}^{-\frac {\left (-3 i \sqrt {7}+1\right ) \left (2 t -3\right )}{16}}\right ) \operatorname {Heaviside}\left (t -\frac {3}{2}\right )\right )}{2688} \end {align*}

Converting the above solution to piecewise it becomes \[ u = \left \{\begin {array}{cc} 0 & t <\frac {3}{2} \\ \left (\frac {i \sqrt {7}}{2688}+\frac {1}{128}\right ) \left (63-3 i \sqrt {7}-32 \,{\mathrm e}^{-\frac {\left (3 i \sqrt {7}+1\right ) \left (2 t -3\right )}{16}}-\left (31-3 i \sqrt {7}\right ) {\mathrm e}^{-\frac {\left (-3 i \sqrt {7}+1\right ) \left (2 t -3\right )}{16}}\right ) & t <\frac {5}{2} \\ \left (\frac {i \sqrt {7}}{2688}+\frac {1}{128}\right ) \left (32 \,{\mathrm e}^{-\frac {\left (3 i \sqrt {7}+1\right ) \left (2 t -5\right )}{16}}+\left (31-3 i \sqrt {7}\right ) {\mathrm e}^{-\frac {\left (-3 i \sqrt {7}+1\right ) \left (2 t -5\right )}{16}}-32 \,{\mathrm e}^{-\frac {\left (3 i \sqrt {7}+1\right ) \left (2 t -3\right )}{16}}-\left (31-3 i \sqrt {7}\right ) {\mathrm e}^{-\frac {\left (-3 i \sqrt {7}+1\right ) \left (2 t -3\right )}{16}}\right ) & \frac {5}{2}\le t \end {array}\right . \] Simplifying the solution gives \[ u = \frac {\left (i \sqrt {7}+21\right ) \left (\left \{\begin {array}{cc} 0 & t <\frac {3}{2} \\ 3 i \sqrt {7}\, {\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}}-3 i \sqrt {7}-32 \,{\mathrm e}^{\frac {3}{16}+\frac {3 i \left (-2 t +3\right ) \sqrt {7}}{16}-\frac {t}{8}}-31 \,{\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}}+63 & t <\frac {5}{2} \\ \left (31-3 i \sqrt {7}\right ) {\mathrm e}^{\frac {3 i \left (2 t -5\right ) \sqrt {7}}{16}-\frac {t}{8}+\frac {5}{16}}+3 i \sqrt {7}\, {\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}}-31 \,{\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}}+32 \,{\mathrm e}^{-\frac {3 i \left (2 t -5\right ) \sqrt {7}}{16}-\frac {t}{8}+\frac {5}{16}}-32 \,{\mathrm e}^{\frac {3}{16}+\frac {3 i \left (-2 t +3\right ) \sqrt {7}}{16}-\frac {t}{8}} & \frac {5}{2}\le t \end {array}\right .\right )}{2688} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} u &= \frac {\left (i \sqrt {7}+21\right ) \left (\left \{\begin {array}{cc} 0 & t <\frac {3}{2} \\ 3 i \sqrt {7}\, {\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}}-3 i \sqrt {7}-32 \,{\mathrm e}^{\frac {3}{16}+\frac {3 i \left (-2 t +3\right ) \sqrt {7}}{16}-\frac {t}{8}}-31 \,{\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}}+63 & t <\frac {5}{2} \\ \left (31-3 i \sqrt {7}\right ) {\mathrm e}^{\frac {3 i \left (2 t -5\right ) \sqrt {7}}{16}-\frac {t}{8}+\frac {5}{16}}+3 i \sqrt {7}\, {\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}}-31 \,{\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}}+32 \,{\mathrm e}^{-\frac {3 i \left (2 t -5\right ) \sqrt {7}}{16}-\frac {t}{8}+\frac {5}{16}}-32 \,{\mathrm e}^{\frac {3}{16}+\frac {3 i \left (-2 t +3\right ) \sqrt {7}}{16}-\frac {t}{8}} & \frac {5}{2}\le t \end {array}\right .\right )}{2688} \\ \end{align*}

Verification of solutions

\[ u = \frac {\left (i \sqrt {7}+21\right ) \left (\left \{\begin {array}{cc} 0 & t <\frac {3}{2} \\ 3 i \sqrt {7}\, {\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}}-3 i \sqrt {7}-32 \,{\mathrm e}^{\frac {3}{16}+\frac {3 i \left (-2 t +3\right ) \sqrt {7}}{16}-\frac {t}{8}}-31 \,{\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}}+63 & t <\frac {5}{2} \\ \left (31-3 i \sqrt {7}\right ) {\mathrm e}^{\frac {3 i \left (2 t -5\right ) \sqrt {7}}{16}-\frac {t}{8}+\frac {5}{16}}+3 i \sqrt {7}\, {\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}}-31 \,{\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}}+32 \,{\mathrm e}^{-\frac {3 i \left (2 t -5\right ) \sqrt {7}}{16}-\frac {t}{8}+\frac {5}{16}}-32 \,{\mathrm e}^{\frac {3}{16}+\frac {3 i \left (-2 t +3\right ) \sqrt {7}}{16}-\frac {t}{8}} & \frac {5}{2}\le t \end {array}\right .\right )}{2688} \] Verified OK.

4.10.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [\frac {d}{d t}u^{\prime }+\frac {u^{\prime }}{4}+u=\frac {\mathit {Heaviside}\left (t -\frac {3}{2}\right )}{2}-\frac {\mathit {Heaviside}\left (t -\frac {5}{2}\right )}{2}, u \left (0\right )=0, u^{\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}u^{\prime } \\ \bullet & {} & \textrm {Characteristic polynomial of homogeneous ODE}\hspace {3pt} \\ {} & {} & r^{2}+\frac {1}{4} r +1=0 \\ \bullet & {} & \textrm {Use quadratic formula to solve for}\hspace {3pt} r \\ {} & {} & r =\frac {\left (-\frac {1}{4}\right )\pm \left (\sqrt {-\frac {63}{16}}\right )}{2} \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (-\frac {1}{8}-\frac {3 \,\mathrm {I} \sqrt {7}}{8}, -\frac {1}{8}+\frac {3 \,\mathrm {I} \sqrt {7}}{8}\right ) \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & u_{1}\left (t \right )={\mathrm e}^{-\frac {t}{8}} \cos \left (\frac {3 \sqrt {7}\, t}{8}\right ) \\ \bullet & {} & \textrm {2nd solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & u_{2}\left (t \right )={\mathrm e}^{-\frac {t}{8}} \sin \left (\frac {3 \sqrt {7}\, t}{8}\right ) \\ \bullet & {} & \textrm {General solution of the ODE}\hspace {3pt} \\ {} & {} & u=c_{1} u_{1}\left (t \right )+c_{2} u_{2}\left (t \right )+u_{p}\left (t \right ) \\ \bullet & {} & \textrm {Substitute in solutions of the homogeneous ODE}\hspace {3pt} \\ {} & {} & u=c_{1} {\mathrm e}^{-\frac {t}{8}} \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )+c_{2} {\mathrm e}^{-\frac {t}{8}} \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )+u_{p}\left (t \right ) \\ \square & {} & \textrm {Find a particular solution}\hspace {3pt} u_{p}\left (t \right )\hspace {3pt}\textrm {of the ODE}\hspace {3pt} \\ {} & \circ & \textrm {Use variation of parameters to find}\hspace {3pt} u_{p}\hspace {3pt}\textrm {here}\hspace {3pt} f \left (t \right )\hspace {3pt}\textrm {is the forcing function}\hspace {3pt} \\ {} & {} & \left [u_{p}\left (t \right )=-u_{1}\left (t \right ) \left (\int \frac {u_{2}\left (t \right ) f \left (t \right )}{W \left (u_{1}\left (t \right ), u_{2}\left (t \right )\right )}d t \right )+u_{2}\left (t \right ) \left (\int \frac {u_{1}\left (t \right ) f \left (t \right )}{W \left (u_{1}\left (t \right ), u_{2}\left (t \right )\right )}d t \right ), f \left (t \right )=\frac {\mathit {Heaviside}\left (t -\frac {3}{2}\right )}{2}-\frac {\mathit {Heaviside}\left (t -\frac {5}{2}\right )}{2}\right ] \\ {} & \circ & \textrm {Wronskian of solutions of the homogeneous equation}\hspace {3pt} \\ {} & {} & W \left (u_{1}\left (t \right ), u_{2}\left (t \right )\right )=\left [\begin {array}{cc} {\mathrm e}^{-\frac {t}{8}} \cos \left (\frac {3 \sqrt {7}\, t}{8}\right ) & {\mathrm e}^{-\frac {t}{8}} \sin \left (\frac {3 \sqrt {7}\, t}{8}\right ) \\ -\frac {{\mathrm e}^{-\frac {t}{8}} \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )}{8}-\frac {3 \,{\mathrm e}^{-\frac {t}{8}} \sqrt {7}\, \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )}{8} & -\frac {{\mathrm e}^{-\frac {t}{8}} \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )}{8}+\frac {3 \,{\mathrm e}^{-\frac {t}{8}} \sqrt {7}\, \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )}{8} \end {array}\right ] \\ {} & \circ & \textrm {Compute Wronskian}\hspace {3pt} \\ {} & {} & W \left (u_{1}\left (t \right ), u_{2}\left (t \right )\right )=\frac {3 \sqrt {7}\, {\mathrm e}^{-\frac {t}{4}}}{8} \\ {} & \circ & \textrm {Substitute functions into equation for}\hspace {3pt} u_{p}\left (t \right ) \\ {} & {} & u_{p}\left (t \right )=-\frac {4 \sqrt {7}\, {\mathrm e}^{-\frac {t}{8}} \left (\cos \left (\frac {3 \sqrt {7}\, t}{8}\right ) \left (\int {\mathrm e}^{\frac {t}{8}} \sin \left (\frac {3 \sqrt {7}\, t}{8}\right ) \left (\mathit {Heaviside}\left (t -\frac {3}{2}\right )-\mathit {Heaviside}\left (t -\frac {5}{2}\right )\right )d t \right )-\sin \left (\frac {3 \sqrt {7}\, t}{8}\right ) \left (\int {\mathrm e}^{\frac {t}{8}} \cos \left (\frac {3 \sqrt {7}\, t}{8}\right ) \left (\mathit {Heaviside}\left (t -\frac {3}{2}\right )-\mathit {Heaviside}\left (t -\frac {5}{2}\right )\right )d t \right )\right )}{21} \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & u_{p}\left (t \right )=\frac {\mathit {Heaviside}\left (t -\frac {3}{2}\right ) \left (\left (\left (-21 \cos \left (\frac {3 \sqrt {7}}{8}\right )+\frac {21}{2}\right ) \cos \left (\frac {3 \sqrt {7}}{16}\right )+\sqrt {7}\, \sin \left (\frac {3 \sqrt {7}}{16}\right ) \left (\cos \left (\frac {3 \sqrt {7}}{8}\right )+\frac {1}{2}\right )\right ) \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )-\left (\sqrt {7}\, \left (\cos \left (\frac {3 \sqrt {7}}{8}\right )-\frac {1}{2}\right ) \cos \left (\frac {3 \sqrt {7}}{16}\right )+21 \sin \left (\frac {3 \sqrt {7}}{16}\right ) \left (\cos \left (\frac {3 \sqrt {7}}{8}\right )+\frac {1}{2}\right )\right ) \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )\right ) {\mathrm e}^{\frac {3}{16}-\frac {t}{8}}}{21}-\frac {\mathit {Heaviside}\left (t -\frac {5}{2}\right ) \left (\left (\sqrt {7}\, \sin \left (\frac {15 \sqrt {7}}{16}\right )+\cos \left (\frac {3 \sqrt {7}}{16}\right ) \left (-21-42 \cos \left (\frac {3 \sqrt {7}}{4}\right )+42 \cos \left (\frac {3 \sqrt {7}}{8}\right )\right )\right ) \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )-\sin \left (\frac {3 \sqrt {7}\, t}{8}\right ) \left (\left (21+42 \cos \left (\frac {3 \sqrt {7}}{4}\right )+42 \cos \left (\frac {3 \sqrt {7}}{8}\right )\right ) \sin \left (\frac {3 \sqrt {7}}{16}\right )+\sqrt {7}\, \cos \left (\frac {15 \sqrt {7}}{16}\right )\right )\right ) {\mathrm e}^{\frac {5}{16}-\frac {t}{8}}}{42}-\frac {\mathit {Heaviside}\left (t -\frac {5}{2}\right )}{2}+\frac {\mathit {Heaviside}\left (t -\frac {3}{2}\right )}{2} \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & u=c_{1} {\mathrm e}^{-\frac {t}{8}} \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )+c_{2} {\mathrm e}^{-\frac {t}{8}} \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )+\frac {\mathit {Heaviside}\left (t -\frac {3}{2}\right ) \left (\left (\left (-21 \cos \left (\frac {3 \sqrt {7}}{8}\right )+\frac {21}{2}\right ) \cos \left (\frac {3 \sqrt {7}}{16}\right )+\sqrt {7}\, \sin \left (\frac {3 \sqrt {7}}{16}\right ) \left (\cos \left (\frac {3 \sqrt {7}}{8}\right )+\frac {1}{2}\right )\right ) \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )-\left (\sqrt {7}\, \left (\cos \left (\frac {3 \sqrt {7}}{8}\right )-\frac {1}{2}\right ) \cos \left (\frac {3 \sqrt {7}}{16}\right )+21 \sin \left (\frac {3 \sqrt {7}}{16}\right ) \left (\cos \left (\frac {3 \sqrt {7}}{8}\right )+\frac {1}{2}\right )\right ) \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )\right ) {\mathrm e}^{\frac {3}{16}-\frac {t}{8}}}{21}-\frac {\mathit {Heaviside}\left (t -\frac {5}{2}\right ) \left (\left (\sqrt {7}\, \sin \left (\frac {15 \sqrt {7}}{16}\right )+\cos \left (\frac {3 \sqrt {7}}{16}\right ) \left (-21-42 \cos \left (\frac {3 \sqrt {7}}{4}\right )+42 \cos \left (\frac {3 \sqrt {7}}{8}\right )\right )\right ) \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )-\sin \left (\frac {3 \sqrt {7}\, t}{8}\right ) \left (\left (21+42 \cos \left (\frac {3 \sqrt {7}}{4}\right )+42 \cos \left (\frac {3 \sqrt {7}}{8}\right )\right ) \sin \left (\frac {3 \sqrt {7}}{16}\right )+\sqrt {7}\, \cos \left (\frac {15 \sqrt {7}}{16}\right )\right )\right ) {\mathrm e}^{\frac {5}{16}-\frac {t}{8}}}{42}-\frac {\mathit {Heaviside}\left (t -\frac {5}{2}\right )}{2}+\frac {\mathit {Heaviside}\left (t -\frac {3}{2}\right )}{2} \\ \square & {} & \textrm {Check validity of solution}\hspace {3pt} u=c_{1} {\mathrm e}^{-\frac {t}{8}} \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )+c_{2} {\mathrm e}^{-\frac {t}{8}} \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )+\frac {\mathit {Heaviside}\left (t -\frac {3}{2}\right ) \left (\left (\left (-21 \cos \left (\frac {3 \sqrt {7}}{8}\right )+\frac {21}{2}\right ) \cos \left (\frac {3 \sqrt {7}}{16}\right )+\sqrt {7}\, \sin \left (\frac {3 \sqrt {7}}{16}\right ) \left (\cos \left (\frac {3 \sqrt {7}}{8}\right )+\frac {1}{2}\right )\right ) \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )-\left (\sqrt {7}\, \left (\cos \left (\frac {3 \sqrt {7}}{8}\right )-\frac {1}{2}\right ) \cos \left (\frac {3 \sqrt {7}}{16}\right )+21 \sin \left (\frac {3 \sqrt {7}}{16}\right ) \left (\cos \left (\frac {3 \sqrt {7}}{8}\right )+\frac {1}{2}\right )\right ) \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )\right ) {\mathrm e}^{\frac {3}{16}-\frac {t}{8}}}{21}-\frac {\mathit {Heaviside}\left (t -\frac {5}{2}\right ) \left (\left (\sqrt {7}\, \sin \left (\frac {15 \sqrt {7}}{16}\right )+\cos \left (\frac {3 \sqrt {7}}{16}\right ) \left (-21-42 \cos \left (\frac {3 \sqrt {7}}{4}\right )+42 \cos \left (\frac {3 \sqrt {7}}{8}\right )\right )\right ) \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )-\sin \left (\frac {3 \sqrt {7}\, t}{8}\right ) \left (\left (21+42 \cos \left (\frac {3 \sqrt {7}}{4}\right )+42 \cos \left (\frac {3 \sqrt {7}}{8}\right )\right ) \sin \left (\frac {3 \sqrt {7}}{16}\right )+\sqrt {7}\, \cos \left (\frac {15 \sqrt {7}}{16}\right )\right )\right ) {\mathrm e}^{\frac {5}{16}-\frac {t}{8}}}{42}-\frac {\mathit {Heaviside}\left (t -\frac {5}{2}\right )}{2}+\frac {\mathit {Heaviside}\left (t -\frac {3}{2}\right )}{2} \\ {} & \circ & \textrm {Use initial condition}\hspace {3pt} u \left (0\right )=0 \\ {} & {} & 0=c_{1} \\ {} & \circ & \textrm {Compute derivative of the solution}\hspace {3pt} \\ {} & {} & u^{\prime }=-\frac {c_{1} {\mathrm e}^{-\frac {t}{8}} \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )}{8}-\frac {3 c_{1} {\mathrm e}^{-\frac {t}{8}} \sqrt {7}\, \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )}{8}-\frac {c_{2} {\mathrm e}^{-\frac {t}{8}} \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )}{8}+\frac {3 c_{2} {\mathrm e}^{-\frac {t}{8}} \sqrt {7}\, \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )}{8}+\frac {\mathit {Dirac}\left (t -\frac {3}{2}\right ) \left (\left (\left (-21 \cos \left (\frac {3 \sqrt {7}}{8}\right )+\frac {21}{2}\right ) \cos \left (\frac {3 \sqrt {7}}{16}\right )+\sqrt {7}\, \sin \left (\frac {3 \sqrt {7}}{16}\right ) \left (\cos \left (\frac {3 \sqrt {7}}{8}\right )+\frac {1}{2}\right )\right ) \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )-\left (\sqrt {7}\, \left (\cos \left (\frac {3 \sqrt {7}}{8}\right )-\frac {1}{2}\right ) \cos \left (\frac {3 \sqrt {7}}{16}\right )+21 \sin \left (\frac {3 \sqrt {7}}{16}\right ) \left (\cos \left (\frac {3 \sqrt {7}}{8}\right )+\frac {1}{2}\right )\right ) \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )\right ) {\mathrm e}^{\frac {3}{16}-\frac {t}{8}}}{21}+\frac {\mathit {Heaviside}\left (t -\frac {3}{2}\right ) \left (-\frac {3 \left (\left (-21 \cos \left (\frac {3 \sqrt {7}}{8}\right )+\frac {21}{2}\right ) \cos \left (\frac {3 \sqrt {7}}{16}\right )+\sqrt {7}\, \sin \left (\frac {3 \sqrt {7}}{16}\right ) \left (\cos \left (\frac {3 \sqrt {7}}{8}\right )+\frac {1}{2}\right )\right ) \sqrt {7}\, \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )}{8}-\frac {3 \left (\sqrt {7}\, \left (\cos \left (\frac {3 \sqrt {7}}{8}\right )-\frac {1}{2}\right ) \cos \left (\frac {3 \sqrt {7}}{16}\right )+21 \sin \left (\frac {3 \sqrt {7}}{16}\right ) \left (\cos \left (\frac {3 \sqrt {7}}{8}\right )+\frac {1}{2}\right )\right ) \sqrt {7}\, \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )}{8}\right ) {\mathrm e}^{\frac {3}{16}-\frac {t}{8}}}{21}-\frac {\mathit {Heaviside}\left (t -\frac {3}{2}\right ) \left (\left (\left (-21 \cos \left (\frac {3 \sqrt {7}}{8}\right )+\frac {21}{2}\right ) \cos \left (\frac {3 \sqrt {7}}{16}\right )+\sqrt {7}\, \sin \left (\frac {3 \sqrt {7}}{16}\right ) \left (\cos \left (\frac {3 \sqrt {7}}{8}\right )+\frac {1}{2}\right )\right ) \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )-\left (\sqrt {7}\, \left (\cos \left (\frac {3 \sqrt {7}}{8}\right )-\frac {1}{2}\right ) \cos \left (\frac {3 \sqrt {7}}{16}\right )+21 \sin \left (\frac {3 \sqrt {7}}{16}\right ) \left (\cos \left (\frac {3 \sqrt {7}}{8}\right )+\frac {1}{2}\right )\right ) \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )\right ) {\mathrm e}^{\frac {3}{16}-\frac {t}{8}}}{168}-\frac {\mathit {Dirac}\left (t -\frac {5}{2}\right ) \left (\left (\sqrt {7}\, \sin \left (\frac {15 \sqrt {7}}{16}\right )+\cos \left (\frac {3 \sqrt {7}}{16}\right ) \left (-21-42 \cos \left (\frac {3 \sqrt {7}}{4}\right )+42 \cos \left (\frac {3 \sqrt {7}}{8}\right )\right )\right ) \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )-\sin \left (\frac {3 \sqrt {7}\, t}{8}\right ) \left (\left (21+42 \cos \left (\frac {3 \sqrt {7}}{4}\right )+42 \cos \left (\frac {3 \sqrt {7}}{8}\right )\right ) \sin \left (\frac {3 \sqrt {7}}{16}\right )+\sqrt {7}\, \cos \left (\frac {15 \sqrt {7}}{16}\right )\right )\right ) {\mathrm e}^{\frac {5}{16}-\frac {t}{8}}}{42}-\frac {\mathit {Heaviside}\left (t -\frac {5}{2}\right ) \left (-\frac {3 \left (\sqrt {7}\, \sin \left (\frac {15 \sqrt {7}}{16}\right )+\cos \left (\frac {3 \sqrt {7}}{16}\right ) \left (-21-42 \cos \left (\frac {3 \sqrt {7}}{4}\right )+42 \cos \left (\frac {3 \sqrt {7}}{8}\right )\right )\right ) \sin \left (\frac {3 \sqrt {7}\, t}{8}\right ) \sqrt {7}}{8}-\frac {3 \sqrt {7}\, \cos \left (\frac {3 \sqrt {7}\, t}{8}\right ) \left (\left (21+42 \cos \left (\frac {3 \sqrt {7}}{4}\right )+42 \cos \left (\frac {3 \sqrt {7}}{8}\right )\right ) \sin \left (\frac {3 \sqrt {7}}{16}\right )+\sqrt {7}\, \cos \left (\frac {15 \sqrt {7}}{16}\right )\right )}{8}\right ) {\mathrm e}^{\frac {5}{16}-\frac {t}{8}}}{42}+\frac {\mathit {Heaviside}\left (t -\frac {5}{2}\right ) \left (\left (\sqrt {7}\, \sin \left (\frac {15 \sqrt {7}}{16}\right )+\cos \left (\frac {3 \sqrt {7}}{16}\right ) \left (-21-42 \cos \left (\frac {3 \sqrt {7}}{4}\right )+42 \cos \left (\frac {3 \sqrt {7}}{8}\right )\right )\right ) \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )-\sin \left (\frac {3 \sqrt {7}\, t}{8}\right ) \left (\left (21+42 \cos \left (\frac {3 \sqrt {7}}{4}\right )+42 \cos \left (\frac {3 \sqrt {7}}{8}\right )\right ) \sin \left (\frac {3 \sqrt {7}}{16}\right )+\sqrt {7}\, \cos \left (\frac {15 \sqrt {7}}{16}\right )\right )\right ) {\mathrm e}^{\frac {5}{16}-\frac {t}{8}}}{336}-\frac {\mathit {Dirac}\left (t -\frac {5}{2}\right )}{2}+\frac {\mathit {Dirac}\left (t -\frac {3}{2}\right )}{2} \\ {} & \circ & \textrm {Use the initial condition}\hspace {3pt} u^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=0 \\ {} & {} & 0=-\frac {c_{1}}{8}+\frac {3 c_{2} \sqrt {7}}{8} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} c_{1} \hspace {3pt}\textrm {and}\hspace {3pt} c_{2} \\ {} & {} & \left \{c_{1} =0, c_{2} =0\right \} \\ {} & \circ & \textrm {Substitute constant values into general solution and simplify}\hspace {3pt} \\ {} & {} & u=\frac {\mathit {Heaviside}\left (t -\frac {3}{2}\right ) \left (\left (\left (-21 \cos \left (\frac {3 \sqrt {7}}{8}\right )+\frac {21}{2}\right ) \cos \left (\frac {3 \sqrt {7}}{16}\right )+\sqrt {7}\, \sin \left (\frac {3 \sqrt {7}}{16}\right ) \left (\cos \left (\frac {3 \sqrt {7}}{8}\right )+\frac {1}{2}\right )\right ) \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )-\left (\sqrt {7}\, \left (\cos \left (\frac {3 \sqrt {7}}{8}\right )-\frac {1}{2}\right ) \cos \left (\frac {3 \sqrt {7}}{16}\right )+21 \sin \left (\frac {3 \sqrt {7}}{16}\right ) \left (\cos \left (\frac {3 \sqrt {7}}{8}\right )+\frac {1}{2}\right )\right ) \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )\right ) {\mathrm e}^{\frac {3}{16}-\frac {t}{8}}}{21}-\frac {\mathit {Heaviside}\left (t -\frac {5}{2}\right ) \left (\left (\sqrt {7}\, \sin \left (\frac {15 \sqrt {7}}{16}\right )+\cos \left (\frac {3 \sqrt {7}}{16}\right ) \left (-21-42 \cos \left (\frac {3 \sqrt {7}}{4}\right )+42 \cos \left (\frac {3 \sqrt {7}}{8}\right )\right )\right ) \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )-\sin \left (\frac {3 \sqrt {7}\, t}{8}\right ) \left (\left (21+42 \cos \left (\frac {3 \sqrt {7}}{4}\right )+42 \cos \left (\frac {3 \sqrt {7}}{8}\right )\right ) \sin \left (\frac {3 \sqrt {7}}{16}\right )+\sqrt {7}\, \cos \left (\frac {15 \sqrt {7}}{16}\right )\right )\right ) {\mathrm e}^{\frac {5}{16}-\frac {t}{8}}}{42}-\frac {\mathit {Heaviside}\left (t -\frac {5}{2}\right )}{2}+\frac {\mathit {Heaviside}\left (t -\frac {3}{2}\right )}{2} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & u=\frac {\mathit {Heaviside}\left (t -\frac {3}{2}\right ) \left (\left (\left (-21 \cos \left (\frac {3 \sqrt {7}}{8}\right )+\frac {21}{2}\right ) \cos \left (\frac {3 \sqrt {7}}{16}\right )+\sqrt {7}\, \sin \left (\frac {3 \sqrt {7}}{16}\right ) \left (\cos \left (\frac {3 \sqrt {7}}{8}\right )+\frac {1}{2}\right )\right ) \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )-\left (\sqrt {7}\, \left (\cos \left (\frac {3 \sqrt {7}}{8}\right )-\frac {1}{2}\right ) \cos \left (\frac {3 \sqrt {7}}{16}\right )+21 \sin \left (\frac {3 \sqrt {7}}{16}\right ) \left (\cos \left (\frac {3 \sqrt {7}}{8}\right )+\frac {1}{2}\right )\right ) \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )\right ) {\mathrm e}^{\frac {3}{16}-\frac {t}{8}}}{21}-\frac {\mathit {Heaviside}\left (t -\frac {5}{2}\right ) \left (\left (\sqrt {7}\, \sin \left (\frac {15 \sqrt {7}}{16}\right )+\cos \left (\frac {3 \sqrt {7}}{16}\right ) \left (-21-42 \cos \left (\frac {3 \sqrt {7}}{4}\right )+42 \cos \left (\frac {3 \sqrt {7}}{8}\right )\right )\right ) \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )-\sin \left (\frac {3 \sqrt {7}\, t}{8}\right ) \left (\left (21+42 \cos \left (\frac {3 \sqrt {7}}{4}\right )+42 \cos \left (\frac {3 \sqrt {7}}{8}\right )\right ) \sin \left (\frac {3 \sqrt {7}}{16}\right )+\sqrt {7}\, \cos \left (\frac {15 \sqrt {7}}{16}\right )\right )\right ) {\mathrm e}^{\frac {5}{16}-\frac {t}{8}}}{42}-\frac {\mathit {Heaviside}\left (t -\frac {5}{2}\right )}{2}+\frac {\mathit {Heaviside}\left (t -\frac {3}{2}\right )}{2} \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: 0.422 (sec). Leaf size: 128

dsolve([diff(u(t),t$2)+1/4*diff(u(t),t)+u(t)=1/2*(Heaviside(t-3/2)-Heaviside(t-5/2)),u(0) = 0, D(u)(0) = 0],u(t), singsol=all)
 

\[ u \left (t \right ) = \frac {\left (-i \sqrt {7}+21\right ) \operatorname {Heaviside}\left (t -\frac {5}{2}\right ) {\mathrm e}^{\frac {3 i \sqrt {7}\, \left (2 t -5\right )}{16}-\frac {t}{8}+\frac {5}{16}}}{84}+\frac {\operatorname {Heaviside}\left (t -\frac {5}{2}\right ) {\mathrm e}^{-\frac {3 i \sqrt {7}\, \left (2 t -5\right )}{16}-\frac {t}{8}+\frac {5}{16}} \left (i \sqrt {7}+21\right )}{84}+\frac {\left (-i \sqrt {7}-21\right ) \operatorname {Heaviside}\left (t -\frac {3}{2}\right ) {\mathrm e}^{\frac {3}{16}+\frac {3 i \left (-2 t +3\right ) \sqrt {7}}{16}-\frac {t}{8}}}{84}+\frac {\left (-21+i \sqrt {7}\right ) \operatorname {Heaviside}\left (t -\frac {3}{2}\right ) {\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}}}{84}-\frac {\operatorname {Heaviside}\left (t -\frac {5}{2}\right )}{2}+\frac {\operatorname {Heaviside}\left (t -\frac {3}{2}\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.115 (sec). Leaf size: 190

DSolve[{u''[t]+1/4*u'[t]+u[t]==1/2*(UnitStep[t-3/2]-UnitStep[t-5/2]),{u[0]==0,u'[0]==0}},u[t],t,IncludeSingularSolutions -> True]
 

\[ u(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{42} \left (-21 e^{\frac {3}{16}-\frac {t}{8}} \cos \left (\frac {3}{16} \sqrt {7} (3-2 t)\right )+\sqrt {7} e^{\frac {3}{16}-\frac {t}{8}} \sin \left (\frac {3}{16} \sqrt {7} (3-2 t)\right )+21\right ) & \frac {3}{2}5 \\ \end {array} \\ \end {array} \]