20.23.8 problem Problem 8

Internal problem ID [3980]
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 10.8. page 710
Problem number : Problem 8
Date solved : Sunday, March 30, 2025 at 02:13:37 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+13 y&=\delta \left (t -\frac {\pi }{4}\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.331 (sec). Leaf size: 46

ode:=diff(diff(y(t),t),t)-4*diff(y(t),t)+13*y(t) = Dirac(t-1/4*Pi); 
ic:=y(0) = 3, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = -\frac {{\mathrm e}^{2 t} \left (\sqrt {2}\, {\mathrm e}^{-\frac {\pi }{2}} \left (\sin \left (3 t \right )+\cos \left (3 t \right )\right ) \operatorname {Heaviside}\left (t -\frac {\pi }{4}\right )-18 \cos \left (3 t \right )+12 \sin \left (3 t \right )\right )}{6} \]

✓ Mathematica. Time used: 0.196 (sec). Leaf size: 61

ode=D[y[t],{t,2}]-4*D[y[t],t]+13*y[t]==DiracDelta[t-Pi/4]; 
ic={y[0]==3,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{6} e^{2 t} \left (6 (3 \cos (3 t)-2 \sin (3 t))-\sqrt {2} e^{-\pi /2} \theta (12 t-3 \pi ) (\sin (3 t)+\cos (3 t))\right ) \]

✓ Sympy. Time used: 3.191 (sec). Leaf size: 102

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - pi/4) + 13*y(t) - 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 3, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (- \frac {\int \operatorname {Dirac}{\left (t - \frac {\pi }{4} \right )} e^{- 2 t} \sin {\left (3 t \right )}\, dt}{3} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {\pi }{4} \right )} e^{- 2 t} \sin {\left (3 t \right )}\, dt}{3} + 3\right ) \cos {\left (3 t \right )} + \left (\frac {\int \operatorname {Dirac}{\left (t - \frac {\pi }{4} \right )} e^{- 2 t} \cos {\left (3 t \right )}\, dt}{3} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {\pi }{4} \right )} e^{- 2 t} \cos {\left (3 t \right )}\, dt}{3} - 2\right ) \sin {\left (3 t \right )}\right ) e^{2 t} \]