20.23.10 problem Problem 10

Internal problem ID [3982]
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 10
Date solved : Tuesday, March 04, 2025 at 05:22:07 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 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 )&=5\\ y^{\prime }\left (0\right )&=5 \end{align*}

✓ Maple. Time used: 3.148 (sec). Leaf size: 42

ode:=diff(diff(y(t),t),t)+6*diff(y(t),t)+13*y(t) = Dirac(t-1/4*Pi); 
ic:=y(0) = 5, D(y)(0) = 5; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = -\frac {\operatorname {Heaviside}\left (t -\frac {\pi }{4}\right ) \cos \left (2 t \right ) {\mathrm e}^{\frac {3 \pi }{4}-3 t}}{2}+5 \,{\mathrm e}^{-3 t} \left (\cos \left (2 t \right )+2 \sin \left (2 t \right )\right ) \]

✓ Mathematica. Time used: 0.28 (sec). Leaf size: 121

ode=D[y[t],{t,2}]+46*D[y[t],t]+13*y[t]==DiracDelta[t-Pi/4]; 
ic={y[0]==1,Derivative[1][y][0] ==-1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{516} e^{-2 \sqrt {129} t-23 t-\frac {\sqrt {129} \pi }{2}} \left (2 e^{\frac {\sqrt {129} \pi }{2}} \left (\left (129+11 \sqrt {129}\right ) e^{4 \sqrt {129} t}+129-11 \sqrt {129}\right )-\sqrt {129} e^{23 \pi /4} \left (e^{\sqrt {129} \pi }-e^{4 \sqrt {129} t}\right ) \theta (4 t-\pi )\right ) \]

✓ Sympy. Time used: 5.859 (sec). Leaf size: 102

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - pi/4) + 13*y(t) + 6*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 5, Subs(Derivative(y(t), t), t, 0): 5} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (- \frac {\int \operatorname {Dirac}{\left (t - \frac {\pi }{4} \right )} e^{3 t} \sin {\left (2 t \right )}\, dt}{2} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {\pi }{4} \right )} e^{3 t} \sin {\left (2 t \right )}\, dt}{2} + 5\right ) \cos {\left (2 t \right )} + \left (\frac {\int \operatorname {Dirac}{\left (t - \frac {\pi }{4} \right )} e^{3 t} \cos {\left (2 t \right )}\, dt}{2} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {\pi }{4} \right )} e^{3 t} \cos {\left (2 t \right )}\, dt}{2} + 10\right ) \sin {\left (2 t \right )}\right ) e^{- 3 t} \]