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 : Monday, January 27, 2025 at 08:05:00 AM
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*}

✓ Solution by Maple

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

dsolve([diff(y(t),t)-y(t)=4*Heaviside(t-Pi/4)*cos(t-Pi/4),y(0) = 1],y(t), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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