11.4.11 problem 12
Internal
problem
ID
[1505]
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
Date
solved
:
Monday, January 27, 2025 at 04:58:19 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} 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} \end{align*}
Using Laplace method With initial conditions
\begin{align*} u \left (0\right )&=0\\ u^{\prime }\left (0\right )&=0 \end{align*}
✓ Solution by Maple
Time used: 2.170 (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)
\[
u = \frac {-21 \left (\cos \left (\frac {3 \sqrt {7}\, \left (-t +5+k \right )}{8}\right )+\frac {31 \sqrt {7}\, \sin \left (\frac {3 \sqrt {7}\, \left (-t +5+k \right )}{8}\right )}{21}\right ) \left (\operatorname {Heaviside}\left (5+k \right )+\operatorname {Heaviside}\left (t -5-k \right )-1\right ) {\mathrm e}^{-\frac {t}{8}+\frac {5}{8}+\frac {k}{8}}+21 \,{\mathrm e}^{\frac {5}{8}-\frac {t}{8}} \operatorname {Heaviside}\left (t -5\right ) \cos \left (\frac {3 \sqrt {7}\, \left (t -5\right )}{8}\right )-31 \sqrt {7}\, {\mathrm e}^{\frac {5}{8}-\frac {t}{8}} \operatorname {Heaviside}\left (t -5\right ) \sin \left (\frac {3 \sqrt {7}\, \left (t -5\right )}{8}\right )+\left (84 k -84 t +441\right ) \operatorname {Heaviside}\left (t -5-k \right )+84 \left (-1+\operatorname {Heaviside}\left (5+k \right )\right ) \left (k +\frac {21}{4}\right ) {\mathrm e}^{-\frac {t}{8}} \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )+4 \sqrt {7}\, \left (k -\frac {11}{4}\right ) \left (-1+\operatorname {Heaviside}\left (5+k \right )\right ) {\mathrm e}^{-\frac {t}{8}} \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )+\left (84 t -441\right ) \operatorname {Heaviside}\left (t -5\right )}{84 k}
\]
✓ Solution by Mathematica
Time used: 12.512 (sec). Leaf size: 486
DSolve[{D[u[t],{t,2}]+1/4*D[u[t],t]+u[t]==1/k*(UnitStep[t-5]*(t-5)-UnitStep[t-(5+k)]*(t-(5+k)) ),{u[0]==0,Derivative[1][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*}