72.21.2 problem 2

Internal problem ID [14901]
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 : 2
Date solved : Monday, March 31, 2025 at 01:01:58 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+5 y&=\operatorname {Heaviside}\left (t -2\right ) \sin \left (4 t -8\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=-2\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.574 (sec). Leaf size: 100

ode:=diff(diff(y(t),t),t)+diff(y(t),t)+5*y(t) = Heaviside(t-2)*sin(4*t-8); 
ic:=y(0) = -2, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {4 \cos \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right ) {\mathrm e}^{1-\frac {t}{2}} \operatorname {Heaviside}\left (t -2\right )}{137}+\frac {92 \sin \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right ) {\mathrm e}^{1-\frac {t}{2}} \operatorname {Heaviside}\left (t -2\right ) \sqrt {19}}{2603}-2 \,{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {19}\, t}{2}\right )-\frac {2 \,{\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {19}\, t}{2}\right ) \sqrt {19}}{19}-\frac {4 \left (\cos \left (4 t -8\right )+\frac {11 \sin \left (4 t -8\right )}{4}\right ) \operatorname {Heaviside}\left (t -2\right )}{137} \]

✓ Mathematica. Time used: 10.086 (sec). Leaf size: 877

ode=D[y[t],{t,2}]+D[y[t],t]+5*y[t]==UnitStep[t-2]*Sin[4*(t-2)]; 
ic={y[0]==-2,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(5*y(t) - sin(4*t - 8)*Heaviside(t - 2) + Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): -2, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

Timed Out