62.3.6 problem 6.11
Internal
problem
ID
[15440]
Book
:
Differential
Equations,
Linear,
Nonlinear,
Ordinary,
Partial.
A.C.
King,
J.Billingham,
S.R.Otto.
Cambridge
Univ.
Press
2003
Section
:
Chapter
6.
Laplace
transforms.
Problems
page
172
Problem
number
:
6.11
Date
solved
:
Thursday, October 02, 2025 at 10:14:11 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} c v^{\prime \prime }+\frac {v^{\prime }}{r}+\frac {v}{L}&=\delta \left (t -1\right )-\delta \left (t \right ) \end{align*}
Using Laplace method
✓ Maple. Time used: 0.175 (sec). Leaf size: 272
ode:=c*diff(diff(v(t),t),t)+1/r*diff(v(t),t)+v(t)/L = Dirac(t-1)-Dirac(t);
dsolve(ode,v(t),method='laplace');
\[
v = {\mathrm e}^{-\frac {t}{2 r c}} \cosh \left (\frac {t \sqrt {L \left (-4 r^{2} c +L \right )}}{2 L r c}\right ) v \left (0\right )+\frac {\left (\left (-{\mathrm e}^{-\frac {-1+t -\frac {\sqrt {L \left (-4 r^{2} c +L \right )}}{L}}{2 r c}}+{\mathrm e}^{-\frac {-1+t +\frac {\sqrt {L \left (-4 r^{2} c +L \right )}}{L}}{2 r c}}\right ) \cosh \left (\frac {t \sqrt {L \left (-4 r^{2} c +L \right )}}{2 L r c}\right ) \operatorname {Heaviside}\left (t -1\right ) r +\left ({\mathrm e}^{-\frac {t}{2 r c}} \left (2 v^{\prime }\left (0\right ) c r -2 r +v \left (0\right )\right )+r \left ({\mathrm e}^{-\frac {-1+t +\frac {\sqrt {L \left (-4 r^{2} c +L \right )}}{L}}{2 r c}}+{\mathrm e}^{-\frac {-1+t -\frac {\sqrt {L \left (-4 r^{2} c +L \right )}}{L}}{2 r c}}\right ) \operatorname {Heaviside}\left (t -1\right )\right ) \sinh \left (\frac {t \sqrt {L \left (-4 r^{2} c +L \right )}}{2 L r c}\right )\right ) \sqrt {L \left (-4 r^{2} c +L \right )}}{-4 r^{2} c +L}
\]
✓ Mathematica. Time used: 0.077 (sec). Leaf size: 72
ode=c*D[v[t],{t,2}]+1/r*D[v[t],t]+v[t]/L==D[UnitStep[t-1]-UnitStep[t] ,t];
ic={};
DSolve[{ode,ic},v[t],t,IncludeSingularSolutions->True]
\begin{align*} v(t)&\to e^{-\frac {\frac {t \sqrt {L-4 c r^2}}{\sqrt {L}}+t}{2 c r}} \left (c_2 e^{\frac {t \sqrt {L-4 c r^2}}{c \sqrt {L} r}}+c_1\right ) \end{align*}
✓ Sympy. Time used: 0.914 (sec). Leaf size: 284
from sympy import *
t = symbols("t")
c = symbols("c")
r = symbols("r")
L = symbols("L")
v = Function("v")
ode = Eq(c*Derivative(v(t), (t, 2)) - Derivative(-Heaviside(t) + Heaviside(t - 1), t) + Derivative(v(t), t)/r + v(t)/L,0)
ics = {}
dsolve(ode,func=v(t),ics=ics)
\[
v{\left (t \right )} = C_{1} e^{\frac {t \left (-1 + \frac {\sqrt {L \left (L - 4 c r^{2}\right )}}{L}\right )}{2 c r}} + C_{2} e^{- \frac {t \left (1 + \frac {\sqrt {L \left (L - 4 c r^{2}\right )}}{L}\right )}{2 c r}} - \frac {L r e^{\frac {- t + 1 - \frac {t \sqrt {L \left (L - 4 c r^{2}\right )}}{L} + \frac {\sqrt {L \left (L - 4 c r^{2}\right )}}{L}}{2 c r}} \theta \left (t - 1\right )}{\sqrt {L \left (L - 4 c r^{2}\right )}} + \frac {L r e^{\frac {- t + 1 + \frac {t \sqrt {L \left (L - 4 c r^{2}\right )}}{L} - \frac {\sqrt {L \left (L - 4 c r^{2}\right )}}{L}}{2 c r}} \theta \left (t - 1\right )}{\sqrt {L \left (L - 4 c r^{2}\right )}} - \frac {L r e^{\frac {t \left (-1 + \frac {\sqrt {L \left (L - 4 c r^{2}\right )}}{L}\right )}{2 c r}} \theta \left (t\right )}{\sqrt {L \left (L - 4 c r^{2}\right )}} + \frac {L r e^{- \frac {t \left (1 + \frac {\sqrt {L \left (L - 4 c r^{2}\right )}}{L}\right )}{2 c r}} \theta \left (t\right )}{\sqrt {L \left (L - 4 c r^{2}\right )}}
\]