76.20.15 problem 16 (c.2)
Internal
problem
ID
[17763]
Book
:
Differential
equations.
An
introduction
to
modern
methods
and
applications.
James
Brannan,
William
E.
Boyce.
Third
edition.
Wiley
2015
Section
:
Chapter
5.
The
Laplace
transform.
Section
5.6
(Differential
equations
with
Discontinuous
Forcing
Functions).
Problems
at
page
342
Problem
number
:
16
(c.2)
Date
solved
:
Tuesday, January 28, 2025 at 11:02:32 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} u^{\prime \prime }+\frac {u^{\prime }}{4}+u&=\left \{\begin {array}{cc} 1 & \frac {3}{2}\le t <\frac {5}{2} \\ 0 & \operatorname {otherwise} \end {array}\right . \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: 17.909 (sec). Leaf size: 184
dsolve([diff(u(t),t$2)+1/4*diff(u(t),t)+u(t)=piecewise(3/2<=t and t<5/2, 1, true ,0),u(0) = 0, D(u)(0) = 0],u(t), singsol=all)
\[
u = \frac {\left (i \sqrt {7}+21\right ) \left (\left \{\begin {array}{cc} 0 & t <\frac {3}{2} \\ 3 i {\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}} \sqrt {7}-3 i \sqrt {7}-32 \,{\mathrm e}^{\frac {3}{16}+\frac {3 i \left (3-2 t \right ) \sqrt {7}}{16}-\frac {t}{8}}-31 \,{\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}}+63 & t <\frac {5}{2} \\ \left (31-3 i \sqrt {7}\right ) {\mathrm e}^{\frac {3 i \sqrt {7}\, \left (-5+2 t \right )}{16}+\frac {5}{16}-\frac {t}{8}}+3 i {\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}} \sqrt {7}-31 \,{\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}}+32 \,{\mathrm e}^{-\frac {3 i \sqrt {7}\, \left (-5+2 t \right )}{16}+\frac {5}{16}-\frac {t}{8}}-32 \,{\mathrm e}^{\frac {3}{16}+\frac {3 i \left (3-2 t \right ) \sqrt {7}}{16}-\frac {t}{8}} & \frac {5}{2}\le t \end {array}\right .\right )}{1344}
\]
✓ Solution by Mathematica
Time used: 0.105 (sec). Leaf size: 197
DSolve[{D[u[t],{t,2}]+1/4*D[u[t],t]+u[t]==Piecewise[{ {1,3/2<= t <5/2}, {0,True}}],{u[0]==0,Derivative[1][u][0] ==0}},u[t],t,IncludeSingularSolutions -> True]
\[
u(t)\to \begin {array}{cc} \{ & \begin {array}{cc} -e^{\frac {3}{16}-\frac {t}{8}} \cos \left (\frac {3}{16} \sqrt {7} (3-2 t)\right )+\frac {e^{\frac {3}{16}-\frac {t}{8}} \sin \left (\frac {3}{16} \sqrt {7} (3-2 t)\right )}{3 \sqrt {7}}+1 & \frac {3}{2}<t\leq \frac {5}{2} \\ \frac {e^{\frac {3}{16}-\frac {t}{8}} \left (-3 \sqrt {7} \cos \left (\frac {3}{16} \sqrt {7} (3-2 t)\right )+3 \sqrt {7} \sqrt [8]{e} \cos \left (\frac {3}{16} \sqrt {7} (5-2 t)\right )+\sin \left (\frac {3}{16} \sqrt {7} (3-2 t)\right )-\sqrt [8]{e} \sin \left (\frac {3}{16} \sqrt {7} (5-2 t)\right )\right )}{3 \sqrt {7}} & 2 t>5 \\ \end {array} \\ \end {array}
\]