76.21.17 problem 15 (b)

Internal problem ID [17710]
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 : 15 (b)
Date solved : Monday, March 31, 2025 at 04:25:38 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+\frac {y^{\prime }}{10}+y&=k \delta \left (t -1\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: 0.239 (sec). Leaf size: 29

ode:=diff(diff(y(t),t),t)+1/10*diff(y(t),t)+y(t) = k*Dirac(t-1); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {20 k \sqrt {399}\, \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{\frac {1}{20}-\frac {t}{20}} \sin \left (\frac {\sqrt {399}\, \left (t -1\right )}{20}\right )}{399} \]

✓ Mathematica. Time used: 0.099 (sec). Leaf size: 41

ode=D[y[t],{t,2}]+1/10*D[y[t],t]+y[t]==k*DiracDelta[t-1]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {20 k e^{\frac {1-t}{20}} \theta (t-1) \sin \left (\frac {1}{20} \sqrt {399} (t-1)\right )}{\sqrt {399}} \]

✓ Sympy. Time used: 2.909 (sec). Leaf size: 156

from sympy import * 
t = symbols("t") 
k = symbols("k") 
y = Function("y") 
ode = Eq(-k*Dirac(t - 1) + y(t) + Derivative(y(t), t)/10 + 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)
 

\[ y{\left (t \right )} = \left (\left (- \frac {20 \sqrt {399} k \int \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{20}} \sin {\left (\frac {\sqrt {399} t}{20} \right )}\, dt}{399} + \frac {20 \sqrt {399} k \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{20}} \sin {\left (\frac {\sqrt {399} t}{20} \right )}\, dt}{399}\right ) \cos {\left (\frac {\sqrt {399} t}{20} \right )} + \left (\frac {20 \sqrt {399} k \int \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{20}} \cos {\left (\frac {\sqrt {399} t}{20} \right )}\, dt}{399} - \frac {20 \sqrt {399} k \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{20}} \cos {\left (\frac {\sqrt {399} t}{20} \right )}\, dt}{399}\right ) \sin {\left (\frac {\sqrt {399} t}{20} \right )}\right ) e^{- \frac {t}{20}} \]