70.21.10 problem 10

Internal problem ID [19060]
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 : Thursday, October 02, 2025 at 03:37:36 PM
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*}

✓ Maple. Time used: 0.157 (sec). Leaf size: 34

ode:=2*diff(diff(y(t),t),t)+diff(y(t),t)+6*y(t) = Dirac(t-1/6*Pi)*sin(t); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 1.06 (sec). Leaf size: 221

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

\begin{align*} y(t)&\to -e^{-t/4} \left (\sin \left (\frac {\sqrt {47} t}{4}\right ) \int _1^0\frac {6 e^{\pi /24} \cos \left (\frac {\sqrt {47} \pi }{24}\right ) \delta (\pi -6 K[1])}{\sqrt {47}}dK[1]-\sin \left (\frac {\sqrt {47} t}{4}\right ) \int _1^t\frac {6 e^{\pi /24} \cos \left (\frac {\sqrt {47} \pi }{24}\right ) \delta (\pi -6 K[1])}{\sqrt {47}}dK[1]+\cos \left (\frac {\sqrt {47} t}{4}\right ) \int _1^0-\frac {6 e^{\pi /24} \delta (\pi -6 K[2]) \sin \left (\frac {\sqrt {47} \pi }{24}\right )}{\sqrt {47}}dK[2]-\cos \left (\frac {\sqrt {47} t}{4}\right ) \int _1^t-\frac {6 e^{\pi /24} \delta (\pi -6 K[2]) \sin \left (\frac {\sqrt {47} \pi }{24}\right )}{\sqrt {47}}dK[2]\right ) \end{align*}

✓ Sympy. Time used: 10.236 (sec). Leaf size: 170

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - pi/6)*sin(t) + 6*y(t) + Derivative(y(t), t) + 2*Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (- \frac {2 \sqrt {47} \int \operatorname {Dirac}{\left (t - \frac {\pi }{6} \right )} e^{\frac {t}{4}} \sin {\left (t \right )} \sin {\left (\frac {\sqrt {47} t}{4} \right )}\, dt}{47} + \frac {2 \sqrt {47} \int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {\pi }{6} \right )} e^{\frac {t}{4}} \sin {\left (t \right )} \sin {\left (\frac {\sqrt {47} t}{4} \right )}\, dt}{47}\right ) \cos {\left (\frac {\sqrt {47} t}{4} \right )} + \left (\frac {2 \sqrt {47} \int \operatorname {Dirac}{\left (t - \frac {\pi }{6} \right )} e^{\frac {t}{4}} \sin {\left (t \right )} \cos {\left (\frac {\sqrt {47} t}{4} \right )}\, dt}{47} - \frac {2 \sqrt {47} \int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {\pi }{6} \right )} e^{\frac {t}{4}} \sin {\left (t \right )} \cos {\left (\frac {\sqrt {47} t}{4} \right )}\, dt}{47}\right ) \sin {\left (\frac {\sqrt {47} t}{4} \right )}\right ) e^{- \frac {t}{4}} \]