11.4.10 problem 11(c) k=1/2
Internal
problem
ID
[1504]
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
Date
solved
:
Tuesday, March 04, 2025 at 12:37:13 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} u^{\prime \prime }+\frac {u^{\prime }}{4}+u&=\frac {\operatorname {Heaviside}\left (t -\frac {3}{2}\right )}{2}-\frac {\operatorname {Heaviside}\left (t -\frac {5}{2}\right )}{2} \end{align*}
Using Laplace method With initial conditions
\begin{align*} u \left (0\right )&=0\\ u^{\prime }\left (0\right )&=0 \end{align*}
✓ Maple. Time used: 0.457 (sec). Leaf size: 132
ode:=diff(diff(u(t),t),t)+1/4*diff(u(t),t)+u(t) = 1/2*Heaviside(t-3/2)-1/2*Heaviside(t-5/2);
ic:=u(0) = 0, D(u)(0) = 0;
dsolve([ode,ic],u(t),method='laplace');
\[
u = \frac {\left (-i \sqrt {7}+21\right ) \operatorname {Heaviside}\left (t -\frac {5}{2}\right ) {\mathrm e}^{\frac {3 i \sqrt {7}\, \left (2 t -5\right )}{16}-\frac {t}{8}+\frac {5}{16}}}{84}+\frac {\operatorname {Heaviside}\left (t -\frac {5}{2}\right ) {\mathrm e}^{-\frac {3 i \sqrt {7}\, \left (2 t -5\right )}{16}-\frac {t}{8}+\frac {5}{16}} \left (i \sqrt {7}+21\right )}{84}+\frac {\left (-i \sqrt {7}-21\right ) \operatorname {Heaviside}\left (t -\frac {3}{2}\right ) {\mathrm e}^{\frac {3}{16}+\frac {3 i \left (-2 t +3\right ) \sqrt {7}}{16}-\frac {t}{8}}}{84}+\frac {\left (-21+i \sqrt {7}\right ) \operatorname {Heaviside}\left (t -\frac {3}{2}\right ) {\mathrm e}^{\frac {\left (3 i \sqrt {7}-1\right ) \left (2 t -3\right )}{16}}}{84}-\frac {\operatorname {Heaviside}\left (t -\frac {5}{2}\right )}{2}+\frac {\operatorname {Heaviside}\left (t -\frac {3}{2}\right )}{2}
\]
✓ Mathematica. Time used: 0.115 (sec). Leaf size: 190
ode=D[u[t],{t,2}]+1/4*D[u[t],t]+u[t]==1/2*(UnitStep[t-3/2]-UnitStep[t-5/2]);
ic={u[0]==0,Derivative[1][u][0]==0};
DSolve[{ode,ic},u[t],t,IncludeSingularSolutions->True]
\[
u(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{42} \left (-21 e^{\frac {3}{16}-\frac {t}{8}} \cos \left (\frac {3}{16} \sqrt {7} (3-2 t)\right )+\sqrt {7} e^{\frac {3}{16}-\frac {t}{8}} \sin \left (\frac {3}{16} \sqrt {7} (3-2 t)\right )+21\right ) & \frac {3}{2}<t\leq \frac {5}{2} \\ \frac {1}{42} e^{\frac {3}{16}-\frac {t}{8}} \left (-21 \cos \left (\frac {3}{16} \sqrt {7} (3-2 t)\right )+21 \sqrt [8]{e} \cos \left (\frac {3}{16} \sqrt {7} (5-2 t)\right )+\sqrt {7} \left (\sin \left (\frac {3}{16} \sqrt {7} (3-2 t)\right )-\sqrt [8]{e} \sin \left (\frac {3}{16} \sqrt {7} (5-2 t)\right )\right )\right ) & 2 t>5 \\ \end {array} \\ \end {array}
\]
✓ Sympy. Time used: 6.672 (sec). Leaf size: 160
from sympy import *
t = symbols("t")
u = Function("u")
ode = Eq(u(t) + Heaviside(t - 5/2)/2 - Heaviside(t - 3/2)/2 + Derivative(u(t), t)/4 + Derivative(u(t), (t, 2)),0)
ics = {u(0): 0, Subs(Derivative(u(t), t), t, 0): 0}
dsolve(ode,func=u(t),ics=ics)
\[
u{\left (t \right )} = \left (\frac {\sqrt {7} e^{\frac {5}{16}} \sin {\left (\frac {3 \sqrt {7} \left (2 t - 5\right )}{16} \right )} \theta \left (t - \frac {5}{2}\right )}{42} - \frac {\sqrt {7} e^{\frac {3}{16}} \sin {\left (\frac {3 \sqrt {7} \left (2 t - 3\right )}{16} \right )} \theta \left (t - \frac {3}{2}\right )}{42} + \frac {e^{\frac {5}{16}} \cos {\left (\frac {3 \sqrt {7} \left (2 t - 5\right )}{16} \right )} \theta \left (t - \frac {5}{2}\right )}{2} - \frac {e^{\frac {3}{16}} \cos {\left (\frac {3 \sqrt {7} \left (2 t - 3\right )}{16} \right )} \theta \left (t - \frac {3}{2}\right )}{2}\right ) e^{- \frac {t}{8}} - \frac {\theta \left (t - \frac {5}{2}\right )}{2} + \frac {\theta \left (t - \frac {3}{2}\right )}{2}
\]