72.19.4 problem 30
Internal
problem
ID
[14891]
Book
:
DIFFERENTIAL
EQUATIONS
by
Paul
Blanchard,
Robert
L.
Devaney,
Glen
R.
Hall.
4th
edition.
Brooks/Cole.
Boston,
USA.
2012
Section
:
Chapter
6.
Laplace
transform.
Section
6.3
page
600
Problem
number
:
30
Date
solved
:
Monday, March 31, 2025 at 01:01:31 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }+6 y^{\prime }+13 y&=13 \operatorname {Heaviside}\left (t -4\right ) \end{align*}
Using Laplace method With initial conditions
\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=1 \end{align*}
✓ Maple. Time used: 0.439 (sec). Leaf size: 45
ode:=diff(diff(y(t),t),t)+6*diff(y(t),t)+13*y(t) = 13*Heaviside(t-4);
ic:=y(0) = 3, D(y)(0) = 1;
dsolve([ode,ic],y(t),method='laplace');
\[
y = \left (\frac {3}{2}+\frac {5 i}{2}\right ) {\mathrm e}^{\left (-3-2 i\right ) t}+\left (-\frac {1}{2}-\frac {3 i}{4}\right ) {\mathrm e}^{\left (-3-2 i\right ) \left (t -4\right )} \operatorname {Heaviside}\left (t -4\right )+\left (-\frac {1}{2}+\frac {3 i}{4}\right ) {\mathrm e}^{\left (-3+2 i\right ) \left (t -4\right )} \operatorname {Heaviside}\left (t -4\right )+\operatorname {Heaviside}\left (t -4\right )+\left (\frac {3}{2}-\frac {5 i}{2}\right ) {\mathrm e}^{\left (-3+2 i\right ) t}
\]
✓ Mathematica. Time used: 0.034 (sec). Leaf size: 82
ode=D[y[t],{t,2}]-4*D[y[t],t]+5*y[t]==UnitStep[t-4];
ic={y[0]==3,Derivative[1][y][0] ==1};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{2 t} (3 \cos (t)-5 \sin (t)) & t\leq 4 \\ -\frac {1}{5} e^{2 t-8} \cos (4-t)+3 e^{2 t} \cos (t)-\frac {2}{5} e^{2 t-8} \sin (4-t)-5 e^{2 t} \sin (t)+\frac {1}{5} & \text {True} \\ \end {array} \\ \end {array}
\]
✓ Sympy. Time used: 2.684 (sec). Leaf size: 99
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(13*y(t) - 13*Heaviside(t - 4) + 6*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0)
ics = {y(0): 3, Subs(Derivative(y(t), t), t, 0): 1}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = \left (2 e^{12} \sin {\left (t \right )} \sin {\left (t - 8 \right )} \theta \left (t - 4\right ) - 3 e^{12} \sin {\left (t \right )} \cos {\left (t - 8 \right )} \theta \left (t - 4\right ) + 5 \sin {\left (2 t \right )} + 3 \cos {\left (2 t \right )} - e^{12} \cos {\left (8 \right )} \theta \left (t - 4\right ) + \frac {3 e^{12} \sin {\left (8 \right )} \theta \left (t - 4\right )}{2}\right ) e^{- 3 t} + \theta \left (t - 4\right )
\]