72.21.3 problem 3

Internal problem ID [14902]
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 : 3
Date solved : Monday, March 31, 2025 at 01:02:01 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+8 y&=\left (1-\operatorname {Heaviside}\left (t -4\right )\right ) \cos \left (t -4\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: 0.606 (sec). Leaf size: 145

ode:=diff(diff(y(t),t),t)+diff(y(t),t)+8*y(t) = (1-Heaviside(t-4))*cos(t-4); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = -\frac {9 \left (\left (\sqrt {31}\, \sin \left (2 \sqrt {31}\right )-\frac {217 \cos \left (2 \sqrt {31}\right )}{9}\right ) \cos \left (\frac {\sqrt {31}\, t}{2}\right )-\frac {217 \sin \left (\frac {\sqrt {31}\, t}{2}\right ) \left (\frac {9 \cos \left (2 \sqrt {31}\right ) \sqrt {31}}{217}+\sin \left (2 \sqrt {31}\right )\right )}{9}\right ) \operatorname {Heaviside}\left (t -4\right ) {\mathrm e}^{2-\frac {t}{2}}}{1550}-\frac {7 \,{\mathrm e}^{-\frac {t}{2}} \left (\cos \left (4\right )-\frac {\sin \left (4\right )}{7}\right ) \cos \left (\frac {\sqrt {31}\, t}{2}\right )}{50}-\frac {9 \,{\mathrm e}^{-\frac {t}{2}} \left (\cos \left (4\right )+\frac {13 \sin \left (4\right )}{9}\right ) \sqrt {31}\, \sin \left (\frac {\sqrt {31}\, t}{2}\right )}{1550}-\frac {7 \left (-1+\operatorname {Heaviside}\left (t -4\right )\right ) \left (\left (\cos \left (t \right )+\frac {\sin \left (t \right )}{7}\right ) \cos \left (4\right )-\frac {\left (\cos \left (t \right )-7 \sin \left (t \right )\right ) \sin \left (4\right )}{7}\right )}{50} \]

✓ Mathematica. Time used: 10.293 (sec). Leaf size: 993

ode=D[y[t],{t,2}]+D[y[t],t]+8*y[t]==(1-UnitStep[t-4])*Cos[t-4]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
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 - 4) - 1)*cos(t - 4) + 8*y(t) + Derivative(y(t), t) + 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)
 

Timed Out