46.8.5 problem 7
Internal
problem
ID
[7376]
Book
:
ADVANCED
ENGINEERING
MATHEMATICS.
ERWIN
KREYSZIG,
HERBERT
KREYSZIG,
EDWARD
J.
NORMINTON.
10th
edition.
John
Wiley
USA.
2011
Section
:
Chapter
6.
Laplace
Transforms.
Problem
set
6.4,
page
230
Problem
number
:
7
Date
solved
:
Wednesday, March 05, 2025 at 04:24:26 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} 4 y^{\prime \prime }+24 y^{\prime }+37 y&=17 \,{\mathrm e}^{-t}+\delta \left (t -\frac {1}{2}\right ) \end{align*}
Using Laplace method With initial conditions
\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=1 \end{align*}
✓ Maple. Time used: 0.493 (sec). Leaf size: 37
ode:=4*diff(diff(y(t),t),t)+24*diff(y(t),t)+37*y(t) = 17*exp(-t)+Dirac(t-1/2);
ic:=y(0) = 1, D(y)(0) = 1;
dsolve([ode,ic],y(t),method='laplace');
\[
y = \frac {\operatorname {Heaviside}\left (t -\frac {1}{2}\right ) {\mathrm e}^{-3 t +\frac {3}{2}} \sin \left (-\frac {1}{4}+\frac {t}{2}\right )}{2}+4 \,{\mathrm e}^{-3 t} \sin \left (\frac {t}{2}\right )+{\mathrm e}^{-t}
\]
✓ Mathematica. Time used: 0.111 (sec). Leaf size: 63
ode=4*D[y[t],{t,2}]+24*D[y[t],t]+27*y[t]==17*Exp[-t]+DiracDelta[t-1/2];
ic={y[0]==1,Derivative[1][y][0] ==1};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to \frac {1}{84} e^{-9 t/2} \left (7 e^{3/4} \left (e^{3 t}-e^{3/2}\right ) \theta (2 t-1)+12 \left (-7 e^{3 t}+17 e^{7 t/2}-3\right )\right )
\]
✓ Sympy. Time used: 41.988 (sec). Leaf size: 143
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(-Dirac(t - 1/2) + 37*y(t) + 24*Derivative(y(t), t) + 4*Derivative(y(t), (t, 2)) - 17*exp(-t),0)
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 1}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = \left (\left (\frac {\int \limits ^{0} 17 e^{2 t} \sin {\left (\frac {t}{2} \right )}\, dt}{2} - \frac {\int \left (\operatorname {Dirac}{\left (t - \frac {1}{2} \right )} e^{t} + 17\right ) e^{2 t} \sin {\left (\frac {t}{2} \right )}\, dt}{2} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {1}{2} \right )} e^{3 t} \sin {\left (\frac {t}{2} \right )}\, dt}{2} + 1\right ) \cos {\left (\frac {t}{2} \right )} + \left (- \frac {\int \limits ^{0} 17 e^{2 t} \cos {\left (\frac {t}{2} \right )}\, dt}{2} + \frac {\int \left (\operatorname {Dirac}{\left (t - \frac {1}{2} \right )} e^{t} + 17\right ) e^{2 t} \cos {\left (\frac {t}{2} \right )}\, dt}{2} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {1}{2} \right )} e^{3 t} \cos {\left (\frac {t}{2} \right )}\, dt}{2} + 8\right ) \sin {\left (\frac {t}{2} \right )}\right ) e^{- 3 t}
\]