67.22.17 problem 31.7 (j)

Internal problem ID [16931]
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 (j)
Date solved : Thursday, October 02, 2025 at 01:40:28 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-12 y^{\prime }+45 y&=\delta \left (t \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: 0.079 (sec). Leaf size: 14

ode:=diff(diff(y(t),t),t)-12*diff(y(t),t)+45*y(t) = Dirac(t); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {{\mathrm e}^{6 t} \sin \left (3 t \right )}{3} \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 47

ode=D[y[t],{t,2}]-12*D[y[t],t]+45*y[t]==DiracDelta[t]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -e^{6 t} \sin (3 t) \left (\int _1^0\frac {\delta (K[1])}{3}dK[1]-\int _1^t\frac {\delta (K[1])}{3}dK[1]\right ) \end{align*}

✓ Sympy. Time used: 2.565 (sec). Leaf size: 85

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t) + 45*y(t) - 12*Derivative(y(t), 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)
 

\[ y{\left (t \right )} = \left (\left (- \frac {\int \operatorname {Dirac}{\left (t \right )} e^{- 6 t} \sin {\left (3 t \right )}\, dt}{3} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t \right )} e^{- 6 t} \sin {\left (3 t \right )}\, dt}{3}\right ) \cos {\left (3 t \right )} + \left (\frac {\int \operatorname {Dirac}{\left (t \right )} e^{- 6 t} \cos {\left (3 t \right )}\, dt}{3} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t \right )} e^{- 6 t} \cos {\left (3 t \right )}\, dt}{3}\right ) \sin {\left (3 t \right )}\right ) e^{6 t} \]