20.22.7 problem Problem 33

Internal problem ID [3962]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 10, The Laplace Transform and Some Elementary Applications. Exercises for 10.7. page 704
Problem number : Problem 33
Date solved : Tuesday, March 04, 2025 at 05:20:09 PM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }-3 y&=-10 \,{\mathrm e}^{-t +a} \sin \left (-2 t +2 a \right ) \operatorname {Heaviside}\left (t -a \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=5 \end{align*}

✓ Maple. Time used: 6.211 (sec). Leaf size: 101

ode:=diff(y(t),t)-3*y(t) = -10*exp(-t+a)*sin(-2*t+2*a)*Heaviside(t-a); 
ic:=y(0) = 5; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \left (\operatorname {Heaviside}\left (t -a \right )+\operatorname {Heaviside}\left (a \right )-1\right ) {\mathrm e}^{3 t -3 a}-{\mathrm e}^{-t +a} \left (\left (\cos \left (2 t \right )+2 \sin \left (2 t \right )\right ) \cos \left (2 a \right )-2 \sin \left (2 a \right ) \left (\cos \left (2 t \right )-\frac {\sin \left (2 t \right )}{2}\right )\right ) \operatorname {Heaviside}\left (t -a \right )-\left (\operatorname {Heaviside}\left (a \right )-1\right ) \left (\cos \left (2 a \right )-2 \sin \left (2 a \right )\right ) {\mathrm e}^{3 t +a}+5 \,{\mathrm e}^{3 t} \]

✓ Mathematica. Time used: 0.469 (sec). Leaf size: 103

ode=D[y[t],t]-3*y[t]==10*Exp[-(t-a)]*Sin[2*(t-a)]*UnitStep[t-a]; 
ic={y[0]==5}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to e^{-3 a-t} \left (e^{4 t} \theta (-a) \left (-2 e^{4 a} \sin (2 a)+e^{4 a} \cos (2 a)-1\right )+\theta (t-a) \left (2 e^{4 a} \sin (2 (a-t))-e^{4 a} \cos (2 (a-t))+e^{4 t}\right )+5 e^{3 a+4 t}\right ) \]

✓ Sympy. Time used: 1.692 (sec). Leaf size: 110

from sympy import * 
t = symbols("t") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-3*y(t) + 10*exp(a - t)*sin(2*a - 2*t)*Heaviside(-a + t) + Derivative(y(t), t),0) 
ics = {y(0): 5} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- 2 e^{a} \sin {\left (2 a \right )} \theta \left (- a\right ) + e^{a} \cos {\left (2 a \right )} \theta \left (- a\right ) + 5 - e^{- 3 a} \theta \left (- a\right )\right ) e^{3 t} + e^{- 3 a + 3 t} \theta \left (- a + t\right ) + 2 e^{a - t} \sin {\left (2 a - 2 t \right )} \theta \left (- a + t\right ) - e^{a - t} \cos {\left (2 a - 2 t \right )} \theta \left (- a + t\right ) \]