Internal problem ID [855]
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: 12.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {u^{\prime \prime }+\frac {u^{\prime }}{4}+u=\frac {\operatorname {Heaviside}\left (t -5\right ) \left (t -5\right )-\operatorname {Heaviside}\left (t -5-k \right ) \left (t -5-k \right )}{k}} \] With initial conditions \begin {align*} [u \left (0\right ) = 0, u^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 1.938 (sec). Leaf size: 216
dsolve([diff(u(t),t$2)+1/4*diff(u(t),t)+u(t)=1/k*(Heaviside(t-5)*(t-5)-Heaviside(t-(5+k))*(t-(5+k)) ),u(0) = 0, D(u)(0) = 0],u(t), singsol=all)
\[ u \left (t \right ) = \frac {-21 \left (\frac {31 \sin \left (\frac {3 \sqrt {7}\, \left (-t +5+k \right )}{8}\right ) \sqrt {7}}{21}+\cos \left (\frac {3 \sqrt {7}\, \left (-t +5+k \right )}{8}\right )\right ) \left (\operatorname {Heaviside}\left (5+k \right )+\operatorname {Heaviside}\left (t -5-k \right )-1\right ) {\mathrm e}^{-\frac {t}{8}+\frac {5}{8}+\frac {k}{8}}+21 \,{\mathrm e}^{\frac {5}{8}-\frac {t}{8}} \operatorname {Heaviside}\left (t -5\right ) \cos \left (\frac {3 \sqrt {7}\, \left (t -5\right )}{8}\right )-31 \sqrt {7}\, {\mathrm e}^{\frac {5}{8}-\frac {t}{8}} \operatorname {Heaviside}\left (t -5\right ) \sin \left (\frac {3 \sqrt {7}\, \left (t -5\right )}{8}\right )+\left (84 k -84 t +441\right ) \operatorname {Heaviside}\left (t -5-k \right )+84 \,{\mathrm e}^{-\frac {t}{8}} \left (-1+\operatorname {Heaviside}\left (5+k \right )\right ) \left (k +\frac {21}{4}\right ) \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )+4 \,{\mathrm e}^{-\frac {t}{8}} \left (-1+\operatorname {Heaviside}\left (5+k \right )\right ) \left (k -\frac {11}{4}\right ) \sqrt {7}\, \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )+\left (84 t -441\right ) \operatorname {Heaviside}\left (t -5\right )}{84 k} \]
✓ Solution by Mathematica
Time used: 13.449 (sec). Leaf size: 486
DSolve[{u''[t]+1/4*u'[t]+u[t]==1/k*(UnitStep[t-5]*(t-5)-UnitStep[t-(5+k)]*(t-(5+k)) ),{u[0]==0,u'[0]==0}},u[t],t,IncludeSingularSolutions -> True]
\begin{align*} u(t)\to \fbox {$\frac {e^{-t/8} \left (21 e^{\frac {k+5}{8}} \cos \left (\frac {3}{8} \sqrt {7} (k-t+5)\right )-84 k \cos \left (\frac {3 \sqrt {7} t}{8}\right )-441 \cos \left (\frac {3 \sqrt {7} t}{8}\right )+31 \sqrt {7} e^{\frac {k+5}{8}} \sin \left (\frac {3}{8} \sqrt {7} (k-t+5)\right )-4 \sqrt {7} k \sin \left (\frac {3 \sqrt {7} t}{8}\right )+11 \sqrt {7} \sin \left (\frac {3 \sqrt {7} t}{8}\right )+\left (21 e^{t/8} (4 t-21)+21 e^{5/8} \cos \left (\frac {3}{8} \sqrt {7} (t-5)\right )-31 \sqrt {7} e^{5/8} \sin \left (\frac {3}{8} \sqrt {7} (t-5)\right )\right ) \theta (t-5)+\left (-21 e^{t/8} (-4 k+4 t-21)-21 e^{\frac {k+5}{8}} \cos \left (\frac {3}{8} \sqrt {7} (k-t+5)\right )-31 \sqrt {7} e^{\frac {k+5}{8}} \sin \left (\frac {3}{8} \sqrt {7} (k-t+5)\right )\right ) \theta (-k+t-5)\right )}{84 k}\text { if }k<-5$} \\ u(t)\to \fbox {$\frac {e^{-t/8} \left (\left (3 \sqrt {7} e^{t/8} (4 t-21)+3 \sqrt {7} e^{5/8} \cos \left (\frac {3}{8} \sqrt {7} (t-5)\right )-31 e^{5/8} \sin \left (\frac {3}{8} \sqrt {7} (t-5)\right )\right ) \theta (t-5)-\left (3 \sqrt {7} e^{t/8} (-4 k+4 t-21)+3 \sqrt {7} e^{\frac {k+5}{8}} \cos \left (\frac {3}{8} \sqrt {7} (k-t+5)\right )+31 e^{\frac {k+5}{8}} \sin \left (\frac {3}{8} \sqrt {7} (k-t+5)\right )\right ) \theta (-k+t-5)\right )}{12 \sqrt {7} k}\text { if }k>-5$} \\ \end{align*}