Internal problem ID [854]
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.
CAS Maple gives this as 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*}
✓ 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]
\[ u(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{42} \left (-21 e^{\frac {3}{16}-\frac {t}{8}} \cos \left (\frac {3}{16} \sqrt {7} (3-2 t)\right )+\sqrt {7} e^{\frac {3}{16}-\frac {t}{8}} \sin \left (\frac {3}{16} \sqrt {7} (3-2 t)\right )+21\right ) & \frac {3}{2}<t\leq \frac {5}{2} \\ \frac {1}{42} e^{\frac {3}{16}-\frac {t}{8}} \left (-21 \cos \left (\frac {3}{16} \sqrt {7} (3-2 t)\right )+21 \sqrt [8]{e} \cos \left (\frac {3}{16} \sqrt {7} (5-2 t)\right )+\sqrt {7} \left (\sin \left (\frac {3}{16} \sqrt {7} (3-2 t)\right )-\sqrt [8]{e} \sin \left (\frac {3}{16} \sqrt {7} (5-2 t)\right )\right )\right ) & 2 t>5 \\ \end {array} \\ \end {array} \]