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]]
Solve \begin {gather*} \boxed {u^{\prime \prime }+\frac {u^{\prime }}{4}+u-\frac {\theta \left (t -\frac {3}{2}\right )}{2}+\frac {\theta \left (t -\frac {5}{2}\right )}{2}=0} \end {gather*} With initial conditions \begin {align*} [u \relax (0) = 0, u^{\prime }\relax (0) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.029 (sec). Leaf size: 115
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)
\[ u \relax (t ) = \frac {\theta \left (t -\frac {3}{2}\right )}{2}-\frac {{\mathrm e}^{-\frac {t}{8}+\frac {3}{16}} \theta \left (t -\frac {3}{2}\right ) \sin \left (\frac {3 \sqrt {7}\, \left (2 t -3\right )}{16}\right ) \sqrt {7}}{42}-\frac {{\mathrm e}^{-\frac {t}{8}+\frac {3}{16}} \theta \left (t -\frac {3}{2}\right ) \cos \left (\frac {3 \sqrt {7}\, \left (2 t -3\right )}{16}\right )}{2}-\frac {\theta \left (t -\frac {5}{2}\right )}{2}+\frac {{\mathrm e}^{-\frac {t}{8}+\frac {5}{16}} \theta \left (t -\frac {5}{2}\right ) \sin \left (\frac {3 \sqrt {7}\, \left (2 t -5\right )}{16}\right ) \sqrt {7}}{42}+\frac {{\mathrm e}^{-\frac {t}{8}+\frac {5}{16}} \theta \left (t -\frac {5}{2}\right ) \cos \left (\frac {3 \sqrt {7}\, \left (2 t -5\right )}{16}\right )}{2} \]
✓ Solution by Mathematica
Time used: 0.068 (sec). Leaf size: 182
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]
\begin{align*} u(t)\to {cc} \{ & {cc} \frac {1}{42} e^{\frac {3}{16}-\frac {t}{8}} \left (\sqrt {7} \sin \left (\frac {3}{16} \sqrt {7} (3-2 t)\right )-21 \cos \left (\frac {3}{16} \sqrt {7} (3-2 t)\right )\right )+\frac {1}{2} & \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{align*}