67.22.12 problem 31.7 (e)

Internal problem ID [16926]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 31. Delta Functions. Additional Exercises. page 572
Problem number : 31.7 (e)
Date solved : Thursday, October 02, 2025 at 01:40:25 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-16 y&=\delta \left (t -10\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: 0.105 (sec). Leaf size: 16

ode:=diff(diff(y(t),t),t)-16*y(t) = Dirac(t-10); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {\operatorname {Heaviside}\left (t -10\right ) \sinh \left (-40+4 t \right )}{4} \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 104

ode=D[y[t],{t,2}]-16*y[t]==DiracDelta[t-10]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -e^{-4 t} \left (e^{8 t} \int _1^0\frac {\delta (K[1]-10)}{8 e^{40}}dK[1]-e^{8 t} \int _1^t\frac {\delta (K[1]-10)}{8 e^{40}}dK[1]-\int _1^t-\frac {1}{8} e^{40} \delta (K[2]-10)dK[2]+\int _1^0-\frac {1}{8} e^{40} \delta (K[2]-10)dK[2]\right ) \end{align*}

✓ Sympy. Time used: 0.434 (sec). Leaf size: 66

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - 10) - 16*y(t) + 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)
 

\[ y{\left (t \right )} = \left (\frac {\int \operatorname {Dirac}{\left (t - 10 \right )} e^{- 4 t}\, dt}{8} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 10 \right )} e^{- 4 t}\, dt}{8}\right ) e^{4 t} + \left (- \frac {\int \operatorname {Dirac}{\left (t - 10 \right )} e^{4 t}\, dt}{8} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 10 \right )} e^{4 t}\, dt}{8}\right ) e^{- 4 t} \]