20.22.3 problem Problem 29

Internal problem ID [3958]
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 29
Date solved : Tuesday, March 04, 2025 at 05:20:03 PM
CAS classification : [[_linear, `class A`]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1 \end{align*}

✓ Maple. Time used: 3.328 (sec). Leaf size: 30

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

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

✓ Mathematica. Time used: 0.105 (sec). Leaf size: 40

ode=D[y[t],t]-y[t]==4*UnitStep[t-Pi/4]*Cos[t-Pi/4]; 
ic={y[0]==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^t & 4 t\leq \pi \\ -2 \sqrt {2} \cos (t)+e^t+2 e^{t-\frac {\pi }{4}} & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 0.882 (sec). Leaf size: 41

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-y(t) - 4*sin(t + pi/4)*Heaviside(t - pi/4) + Derivative(y(t), t),0) 
ics = {y(0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

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