70.21.13 problem 14 (a)
Internal
problem
ID
[19063]
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
5.7
(Impulse
Functions).
Problems
at
page
350
Problem
number
:
14
(a)
Date
solved
:
Thursday, October 02, 2025 at 03:37:39 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }+\frac {y^{\prime }}{2}+y&=\delta \left (t -1\right ) \end{align*}
Using Laplace method With initial conditions
\begin{align*}
y \left (0\right )&=0 \\
y^{\prime }\left (0\right )&=0 \\
\end{align*}
✓ Maple. Time used: 0.207 (sec). Leaf size: 28
ode:=diff(diff(y(t),t),t)+1/2*diff(y(t),t)+y(t) = Dirac(t-1);
ic:=[y(0) = 0, D(y)(0) = 0];
dsolve([ode,op(ic)],y(t),method='laplace');
\[
y = \frac {4 \sqrt {15}\, \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{\frac {1}{4}-\frac {t}{4}} \sin \left (\frac {\sqrt {15}\, \left (t -1\right )}{4}\right )}{15}
\]
✓ Mathematica. Time used: 0.083 (sec). Leaf size: 201
ode=D[y[t],{t,2}]+1/2*D[y[t],t]+y[t]==DiracDelta[t-1];
ic={y[0]==0,Derivative[1][y][0] ==0};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*} y(t)&\to -e^{-t/4} \left (\sin \left (\frac {\sqrt {15} t}{4}\right ) \int _1^0\frac {4 \sqrt [4]{e} \cos \left (\frac {\sqrt {15}}{4}\right ) \delta (K[1]-1)}{\sqrt {15}}dK[1]-\sin \left (\frac {\sqrt {15} t}{4}\right ) \int _1^t\frac {4 \sqrt [4]{e} \cos \left (\frac {\sqrt {15}}{4}\right ) \delta (K[1]-1)}{\sqrt {15}}dK[1]+\cos \left (\frac {\sqrt {15} t}{4}\right ) \int _1^0-\frac {4 \sqrt [4]{e} \delta (K[2]-1) \sin \left (\frac {\sqrt {15}}{4}\right )}{\sqrt {15}}dK[2]-\cos \left (\frac {\sqrt {15} t}{4}\right ) \int _1^t-\frac {4 \sqrt [4]{e} \delta (K[2]-1) \sin \left (\frac {\sqrt {15}}{4}\right )}{\sqrt {15}}dK[2]\right ) \end{align*}
✓ Sympy. Time used: 1.900 (sec). Leaf size: 150
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(-Dirac(t - 1) + y(t) + Derivative(y(t), t)/2 + Derivative(y(t), (t, 2)),0)
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = \left (\left (- \frac {4 \sqrt {15} \int \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{4}} \sin {\left (\frac {\sqrt {15} t}{4} \right )}\, dt}{15} + \frac {4 \sqrt {15} \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{4}} \sin {\left (\frac {\sqrt {15} t}{4} \right )}\, dt}{15}\right ) \cos {\left (\frac {\sqrt {15} t}{4} \right )} + \left (\frac {4 \sqrt {15} \int \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{4}} \cos {\left (\frac {\sqrt {15} t}{4} \right )}\, dt}{15} - \frac {4 \sqrt {15} \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{4}} \cos {\left (\frac {\sqrt {15} t}{4} \right )}\, dt}{15}\right ) \sin {\left (\frac {\sqrt {15} t}{4} \right )}\right ) e^{- \frac {t}{4}}
\]