76.21.10 problem 10

Internal problem ID [17774]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.7 (Impulse Functions). Problems at page 350
Problem number : 10
Date solved : Tuesday, January 28, 2025 at 11:02:48 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 2 y^{\prime \prime }+y^{\prime }+6 y&=\delta \left (t -\frac {\pi }{6}\right ) \sin \left (t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 14.643 (sec). Leaf size: 34

dsolve([2*diff(y(t),t$2)+diff(y(t),t)+6*y(t)=Dirac(t-Pi/6)*sin(t),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = -\frac {\sqrt {47}\, {\mathrm e}^{\frac {\pi }{24}-\frac {t}{4}} \operatorname {Heaviside}\left (t -\frac {\pi }{6}\right ) \sin \left (\frac {\sqrt {47}\, \left (-6 t +\pi \right )}{24}\right )}{47} \]

✓ Solution by Mathematica

Time used: 0.091 (sec). Leaf size: 46

DSolve[{2*D[y[t],{t,2}]+D[y[t],t]+6*y[t]==DiracDelta[t-Pi/6]*Sin[t],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to -\frac {e^{\frac {1}{24} (\pi -6 t)} \theta (6 t-\pi ) \sin \left (\frac {1}{24} \sqrt {47} (\pi -6 t)\right )}{\sqrt {47}} \]