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 : Monday, January 27, 2025 at 08:05:01 AM
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*}

✓ Solution by Maple

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

dsolve([diff(y(t),t)+2*y(t)=Heaviside(t-Pi)*sin(2*t),y(0) = 3],y(t), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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