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]]
Solve \begin {gather*} \boxed {u^{\prime \prime }+\frac {u^{\prime }}{4}+u-\frac {\theta \left (t -5\right ) \left (t -5\right )-\theta \left (t -5-k \right ) \left (t -5-k \right )}{k}=0} \end {gather*} With initial conditions \begin {align*} [u \relax (0) = 0, u^{\prime }\relax (0) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.656 (sec). Leaf size: 182
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)
\[ u \relax (t ) = \frac {-31 \left (\theta \left (5+k \right )+\theta \left (t -5-k \right )-1\right ) \left (\left (\left (\sin \left (\frac {15 \sqrt {7}}{8}\right ) \sqrt {7}+\frac {21 \cos \left (\frac {15 \sqrt {7}}{8}\right )}{31}\right ) \cos \left (\frac {3 \sqrt {7}\, k}{8}\right )+\sin \left (\frac {3 \sqrt {7}\, k}{8}\right ) \left (\cos \left (\frac {15 \sqrt {7}}{8}\right ) \sqrt {7}-\frac {21 \sin \left (\frac {15 \sqrt {7}}{8}\right )}{31}\right )\right ) \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )-\left (\left (\cos \left (\frac {15 \sqrt {7}}{8}\right ) \sqrt {7}-\frac {21 \sin \left (\frac {15 \sqrt {7}}{8}\right )}{31}\right ) \cos \left (\frac {3 \sqrt {7}\, k}{8}\right )-\left (\sin \left (\frac {15 \sqrt {7}}{8}\right ) \sqrt {7}+\frac {21 \cos \left (\frac {15 \sqrt {7}}{8}\right )}{31}\right ) \sin \left (\frac {3 \sqrt {7}\, k}{8}\right )\right ) \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )\right ) {\mathrm e}^{\frac {5}{8}+\frac {k}{8}-\frac {t}{8}}+\left (21 \theta \left (t -5\right ) {\mathrm e}^{\frac {5}{8}-\frac {t}{8}} \cos \left (\frac {15 \sqrt {7}}{8}\right )+31 \sqrt {7}\, \theta \left (t -5\right ) {\mathrm e}^{\frac {5}{8}-\frac {t}{8}} \sin \left (\frac {15 \sqrt {7}}{8}\right )+84 \left (k +\frac {21}{4}\right ) \left (-1+\theta \left (5+k \right )\right ) {\mathrm e}^{-\frac {t}{8}}\right ) \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )+\left (-31 \sqrt {7}\, \theta \left (t -5\right ) {\mathrm e}^{\frac {5}{8}-\frac {t}{8}} \cos \left (\frac {15 \sqrt {7}}{8}\right )+21 \theta \left (t -5\right ) {\mathrm e}^{\frac {5}{8}-\frac {t}{8}} \sin \left (\frac {15 \sqrt {7}}{8}\right )+4 \left (k -\frac {11}{4}\right ) \left (-1+\theta \left (5+k \right )\right ) \sqrt {7}\, {\mathrm e}^{-\frac {t}{8}}\right ) \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )+\left (84 k -84 t +441\right ) \theta \left (t -5-k \right )+\left (84 t -441\right ) \theta \left (t -5\right )}{84 k} \]
✓ Solution by Mathematica
Time used: 5.239 (sec). Leaf size: 436
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 {$\frac {e^{-t/8} \left (-21 (4 k+21) \cos \left (\frac {3 \sqrt {7} t}{8}\right )+e^{\frac {k+5}{8}} \left (21 \cos \left (\frac {3}{8} \sqrt {7} (k-t+5)\right )+31 \sqrt {7} \sin \left (\frac {3}{8} \sqrt {7} (k-t+5)\right )\right )+\sqrt {7} (11-4 k) \sin \left (\frac {3 \sqrt {7} t}{8}\right )+\left (21 e^{t/8} (4 t-21)+e^{5/8} \left (21 \cos \left (\frac {3}{8} \sqrt {7} (t-5)\right )-31 \sqrt {7} \sin \left (\frac {3}{8} \sqrt {7} (t-5)\right )\right )\right ) \theta (t-5)+\left (-21 e^{t/8} (-4 k+4 t-21)-e^{\frac {k+5}{8}} \left (21 \cos \left (\frac {3}{8} \sqrt {7} (k-t+5)\right )+31 \sqrt {7} \sin \left (\frac {3}{8} \sqrt {7} (k-t+5)\right )\right )\right ) \theta (-k+t-5)\right )}{84 k}\text { if }k<-5$} \\ u(t)\to \fbox {$\frac {e^{-t/8} \left (\left (3 \sqrt {7} e^{t/8} (4 t-21)+e^{5/8} \left (3 \sqrt {7} \cos \left (\frac {3}{8} \sqrt {7} (t-5)\right )-31 \sin \left (\frac {3}{8} \sqrt {7} (t-5)\right )\right )\right ) \theta (t-5)+\left (-3 \sqrt {7} e^{t/8} (-4 k+4 t-21)-e^{\frac {k+5}{8}} \left (3 \sqrt {7} \cos \left (\frac {3}{8} \sqrt {7} (k-t+5)\right )+31 \sin \left (\frac {3}{8} \sqrt {7} (k-t+5)\right )\right )\right ) \theta (-k+t-5)\right )}{12 \sqrt {7} k}\text { if }k>-5$} \\ \end{align*}