5.5.9 problem 10(a)

Internal problem ID [1514]
Book : Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section : Chapter 6.5, The Laplace Transform. Impulse functions. page 273
Problem number : 10(a)
Date solved : Tuesday, September 30, 2025 at 04:35:07 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+\frac {y^{\prime }}{2}+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: 0.219 (sec). Leaf size: 28

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

\[ y = \frac {4 \sqrt {15}\, \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{\frac {1}{4}-\frac {t}{4}} \sin \left (\frac {\sqrt {15}\, \left (t -1\right )}{4}\right )}{15} \]

✓ Mathematica. Time used: 0.074 (sec). Leaf size: 40

ode=D[y[t],{t,2}]+1/2*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]
 

\begin{align*} y(t)&\to \frac {4 e^{\frac {1}{4}-\frac {t}{4}} \theta (t-1) \sin \left (\frac {1}{4} \sqrt {15} (t-1)\right )}{\sqrt {15}} \end{align*}

✓ Sympy. Time used: 2.274 (sec). Leaf size: 150

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - 1) + y(t) + Derivative(y(t), t)/2 + 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)
 

\[ y{\left (t \right )} = \left (\left (- \frac {4 \sqrt {15} \int \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{4}} \sin {\left (\frac {\sqrt {15} t}{4} \right )}\, dt}{15} + \frac {4 \sqrt {15} \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{4}} \sin {\left (\frac {\sqrt {15} t}{4} \right )}\, dt}{15}\right ) \cos {\left (\frac {\sqrt {15} t}{4} \right )} + \left (\frac {4 \sqrt {15} \int \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{4}} \cos {\left (\frac {\sqrt {15} t}{4} \right )}\, dt}{15} - \frac {4 \sqrt {15} \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{4}} \cos {\left (\frac {\sqrt {15} t}{4} \right )}\, dt}{15}\right ) \sin {\left (\frac {\sqrt {15} t}{4} \right )}\right ) e^{- \frac {t}{4}} \]