76.21.14 problem 14 (c)

Internal problem ID [17707]
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 : 14 (c)
Date solved : Monday, March 31, 2025 at 04:25:32 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+\frac {y^{\prime }}{4}+y&=\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: 0.144 (sec). Leaf size: 28

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

\[ y = \frac {8 \sqrt {7}\, \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{\frac {1}{8}-\frac {t}{8}} \sin \left (\frac {3 \sqrt {7}\, \left (t -1\right )}{8}\right )}{21} \]

✓ Mathematica. Time used: 0.072 (sec). Leaf size: 42

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

\[ y(t)\to \frac {8 e^{\frac {1}{8}-\frac {t}{8}} \theta (t-1) \sin \left (\frac {3}{8} \sqrt {7} (t-1)\right )}{3 \sqrt {7}} \]

✓ Sympy. Time used: 2.734 (sec). Leaf size: 160

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - 1) + y(t) + Derivative(y(t), t)/4 + 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)
 

\[ y{\left (t \right )} = \left (\left (- \frac {8 \sqrt {7} \int \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{8}} \sin {\left (\frac {3 \sqrt {7} t}{8} \right )}\, dt}{21} + \frac {8 \sqrt {7} \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{8}} \sin {\left (\frac {3 \sqrt {7} t}{8} \right )}\, dt}{21}\right ) \cos {\left (\frac {3 \sqrt {7} t}{8} \right )} + \left (\frac {8 \sqrt {7} \int \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{8}} \cos {\left (\frac {3 \sqrt {7} t}{8} \right )}\, dt}{21} - \frac {8 \sqrt {7} \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{8}} \cos {\left (\frac {3 \sqrt {7} t}{8} \right )}\, dt}{21}\right ) \sin {\left (\frac {3 \sqrt {7} t}{8} \right )}\right ) e^{- \frac {t}{8}} \]