72.21.4 problem 4
Internal
problem
ID
[14895]
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
6.6.
page
624
Problem
number
:
4
Date
solved
:
Thursday, March 13, 2025 at 05:18:54 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }+y^{\prime }+3 y&=\left (1-\operatorname {Heaviside}\left (t -2\right )\right ) {\mathrm e}^{-\frac {t}{10}+\frac {1}{5}} \sin \left (t -2\right ) \end{align*}
Using Laplace method With initial conditions
\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=2 \end{align*}
✓ Maple. Time used: 13.524 (sec). Leaf size: 190
ode:=diff(diff(y(t),t),t)+diff(y(t),t)+3*y(t) = (1-Heaviside(t-2))*exp(-1/10*t+1/5)*sin(t-2);
ic:=y(0) = 1, D(y)(0) = 2;
dsolve([ode,ic],y(t),method='laplace');
\[
y = \frac {8000 \left (\left (\cos \left (t \right )-\frac {191 \sin \left (t \right )}{80}\right ) \cos \left (2\right )+\frac {191 \sin \left (2\right ) \left (\cos \left (t \right )+\frac {80 \sin \left (t \right )}{191}\right )}{80}\right ) \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-\frac {t}{10}+\frac {1}{5}}}{42881}+\frac {100 \left (11 \left (191 \sin \left (2\right )+80 \cos \left (2\right )\right ) \cos \left (\frac {\sqrt {11}\, t}{2}\right )-318 \left (\cos \left (2\right )-\frac {782 \sin \left (2\right )}{795}\right ) \sin \left (\frac {\sqrt {11}\, t}{2}\right ) \sqrt {11}\right ) {\mathrm e}^{\frac {1}{5}-\frac {t}{2}}}{471691}+\left (-\frac {4000}{42881}+\frac {9550 i}{42881}\right ) {\mathrm e}^{\left (-\frac {1}{10}-i\right ) \left (t -2\right )}+\left (-\frac {4000}{42881}-\frac {9550 i}{42881}\right ) {\mathrm e}^{\left (-\frac {1}{10}+i\right ) \left (t -2\right )}+\frac {200 \operatorname {Heaviside}\left (t -2\right ) \left (\left (-159 \sin \left (\sqrt {11}\right ) \sqrt {11}-440 \cos \left (\sqrt {11}\right )\right ) \cos \left (\frac {\sqrt {11}\, t}{2}\right )+\left (159 \sqrt {11}\, \cos \left (\sqrt {11}\right )-440 \sin \left (\sqrt {11}\right )\right ) \sin \left (\frac {\sqrt {11}\, t}{2}\right )\right ) {\mathrm e}^{-\frac {t}{2}+1}}{471691}+\frac {5 \,{\mathrm e}^{-\frac {t}{2}} \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, t}{2}\right )}{11}+{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {11}\, t}{2}\right )
\]
✓ Mathematica. Time used: 13.046 (sec). Leaf size: 1335
ode=D[y[t],{t,2}]+D[y[t],t]+8*y[t]==(1-UnitStep[t-2])*Exp[-(t-2)/10]*Sin[t-2];
ic={y[0]==1,Derivative[1][y][0] ==2};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*} \text {Solution too large to show}\end{align*}
✗ Sympy
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq((Heaviside(t - 2) - 1)*exp(1/5 - t/10)*sin(t - 2) + 3*y(t) + Derivative(y(t), t) + Derivative(y(t), (t, 2)),0)
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 2}
dsolve(ode,func=y(t),ics=ics)
Timed Out