90.28.7 problem 7

Internal problem ID [25429]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 6. Discontinuous Functions and the Laplace Transform. Exercises at page 437
Problem number : 7
Date solved : Friday, October 03, 2025 at 12:01:27 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+3 y&=2 \delta \left (t -2\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.062 (sec). Leaf size: 32

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+3*y(t) = 2*Dirac(t-2); 
ic:=[y(0) = 1, D(y)(0) = -1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{2-t}-\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{6-3 t}+{\mathrm e}^{-t} \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 35

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

\begin{align*} y(t)&\to e^{-3 t} \left (\left (e^{2 t+2}-e^6\right ) \theta (t-2)+e^{2 t}\right ) \end{align*}

✓ Sympy. Time used: 0.601 (sec). Leaf size: 133

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-2*Dirac(t - 2) + 3*y(t) + 5*Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): -1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- \frac {2 \sqrt {15} \int \operatorname {Dirac}{\left (t - 2 \right )} \sin {\left (\frac {\sqrt {15} t}{5} \right )}\, dt}{15} + \frac {2 \sqrt {15} \int \limits ^{0} \operatorname {Dirac}{\left (t - 2 \right )} \sin {\left (\frac {\sqrt {15} t}{5} \right )}\, dt}{15} + 1\right ) \cos {\left (\frac {\sqrt {15} t}{5} \right )} + \left (\frac {2 \sqrt {15} \int \operatorname {Dirac}{\left (t - 2 \right )} \cos {\left (\frac {\sqrt {15} t}{5} \right )}\, dt}{15} - \frac {2 \sqrt {15} \int \limits ^{0} \operatorname {Dirac}{\left (t - 2 \right )} \cos {\left (\frac {\sqrt {15} t}{5} \right )}\, dt}{15} - \frac {\sqrt {15}}{3}\right ) \sin {\left (\frac {\sqrt {15} t}{5} \right )} \]