78.4.7 problem 4.b

Internal problem ID [21035]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 5, Laplace transforms. Problems section 5.7
Problem number : 4.b
Date solved : Thursday, October 02, 2025 at 07:01:38 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+13 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 )&=2 \\ \end{align*}

✓ Maple. Time used: 0.451 (sec). Leaf size: 30

ode:=diff(diff(y(t),t),t)-4*diff(y(t),t)+13*y(t) = Dirac(t-1); 
ic:=[y(0) = 0, D(y)(0) = 2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {{\mathrm e}^{2 t} \left ({\mathrm e}^{-2} \sin \left (-3+3 t \right ) \operatorname {Heaviside}\left (t -1\right )+2 \sin \left (3 t \right )\right )}{3} \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 38

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

\begin{align*} y(t)&\to \frac {1}{3} e^{2 t-2} \left (2 e^2 \sin (3 t)-\theta (t-1) \sin (3-3 t)\right ) \end{align*}

✓ Sympy. Time used: 1.800 (sec). Leaf size: 95

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - 1) + 13*y(t) - 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (- \frac {\int \operatorname {Dirac}{\left (t - 1 \right )} e^{- 2 t} \sin {\left (3 t \right )}\, dt}{3} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{- 2 t} \sin {\left (3 t \right )}\, dt}{3}\right ) \cos {\left (3 t \right )} + \left (\frac {\int \operatorname {Dirac}{\left (t - 1 \right )} e^{- 2 t} \cos {\left (3 t \right )}\, dt}{3} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{- 2 t} \cos {\left (3 t \right )}\, dt}{3} + \frac {2}{3}\right ) \sin {\left (3 t \right )}\right ) e^{2 t} \]