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
:
Saturday, March 29, 2025 at 10:56:51 PM
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*}
✓ Maple. Time used: 1.875 (sec). Leaf size: 229
ode:=diff(diff(u(t),t),t)+1/4*diff(u(t),t)+u(t) = 1/k*(Heaviside(t-5)*(t-5)-Heaviside(t-5-k)*(t-5-k));
ic:=u(0) = 0, D(u)(0) = 0;
dsolve([ode,ic],u(t),method='laplace');
\[
u = \frac {\operatorname {Heaviside}\left (t -5\right ) \left (21+31 i \sqrt {7}\right ) {\mathrm e}^{\frac {3 i \sqrt {7}\, \left (t -5\right )}{8}+\frac {5}{8}-\frac {t}{8}}-31 \left (\operatorname {Heaviside}\left (5+k \right )+\operatorname {Heaviside}\left (t -5-k \right )-1\right ) \left (i \sqrt {7}+\frac {21}{31}\right ) {\mathrm e}^{-\frac {3 \left (-t +5+k \right ) \left (i \sqrt {7}-\frac {1}{3}\right )}{8}}+31 \left (i \sqrt {7}-\frac {21}{31}\right ) \left (\operatorname {Heaviside}\left (5+k \right )+\operatorname {Heaviside}\left (t -5-k \right )-1\right ) {\mathrm e}^{\frac {\left (3 i \sqrt {7}+1\right ) \left (-t +5+k \right )}{8}}-4 \left (\operatorname {Heaviside}\left (5+k \right )-1\right ) \left (\left (i k -\frac {11}{4} i\right ) \sqrt {7}-21 k -\frac {441}{4}\right ) {\mathrm e}^{\frac {3 i \sqrt {7}\, t}{8}-\frac {t}{8}}+4 \left (\operatorname {Heaviside}\left (5+k \right )-1\right ) \left (\left (i k -\frac {11}{4} i\right ) \sqrt {7}+21 k +\frac {441}{4}\right ) {\mathrm e}^{-\frac {3 i \sqrt {7}\, t}{8}-\frac {t}{8}}+\left (-31 i \sqrt {7}+21\right ) \operatorname {Heaviside}\left (t -5\right ) {\mathrm e}^{-\frac {3 \left (t -5\right ) \left (i \sqrt {7}+\frac {1}{3}\right )}{8}}+\left (168 k -168 t +882\right ) \operatorname {Heaviside}\left (t -5-k \right )+\left (168 t -882\right ) \operatorname {Heaviside}\left (t -5\right )}{168 k}
\]
✓ Mathematica. Time used: 12.512 (sec). Leaf size: 486
ode=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)) );
ic={u[0]==0,Derivative[1][u][0]==0};
DSolve[{ode,ic},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*}
✓ Sympy. Time used: 13.593 (sec). Leaf size: 583
from sympy import *
t = symbols("t")
k = symbols("k")
u = Function("u")
ode = Eq(u(t) + Derivative(u(t), t)/4 + Derivative(u(t), (t, 2)) - ((t - 5)*Heaviside(t - 5) - (-k + t - 5)*Heaviside(-k + t - 5))/k,0)
ics = {u(0): 0, Subs(Derivative(u(t), t), t, 0): 0}
dsolve(ode,func=u(t),ics=ics)
\[
\text {Solution too large to show}
\]