20.22.4 problem Problem 30

Internal problem ID [3959]
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 30
Date solved : Tuesday, March 04, 2025 at 05:20:05 PM
CAS classification : [[_linear, `class A`]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=3 \end{align*}

✓ Maple. Time used: 3.270 (sec). Leaf size: 46

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

\[ y = \frac {\operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{-2 t +2 \pi }}{4}+\frac {\operatorname {Heaviside}\left (t -\pi \right ) \left (-\cos \left (2 t \right )+\sin \left (2 t \right )\right )}{4}+3 \,{\mathrm e}^{-2 t} \]

✓ Mathematica. Time used: 0.117 (sec). Leaf size: 55

ode=D[y[t],t]+2*y[t]==UnitStep[t-Pi]*Sin[2*t]; 
ic={y[0]==3}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 3 e^{-2 t} & t\leq \pi \\ \frac {1}{4} e^{-2 t} \left (-e^{2 t} \cos (2 t)+e^{2 t} \sin (2 t)+e^{2 \pi }+12\right ) & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 1.535 (sec). Leaf size: 48

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

\[ y{\left (t \right )} = \frac {e^{- 2 t + 2 \pi } \theta \left (t - \pi \right )}{4} - \frac {\sqrt {2} \cos {\left (2 t + \frac {\pi }{4} \right )} \theta \left (t - \pi \right )}{4} + 3 e^{- 2 t} \]