67.22.10 problem 31.7 (c)

Internal problem ID [16924]
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 (c)
Date solved : Thursday, October 02, 2025 at 01:40:24 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }&=\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 )&=1 \\ \end{align*}

✓ Maple. Time used: 0.120 (sec). Leaf size: 30

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

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

✓ Mathematica. Time used: 36.327 (sec). Leaf size: 126

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

\begin{align*} y(t)&\to \int _1^t\left (e^{-3 K[2]} \left (1-\int _1^0e^3 \delta (K[1]-1)dK[1]\right )+e^{-3 K[2]} \int _1^{K[2]}e^3 \delta (K[1]-1)dK[1]\right )dK[2]-\int _1^0\left (e^{-3 K[2]} \left (1-\int _1^0e^3 \delta (K[1]-1)dK[1]\right )+e^{-3 K[2]} \int _1^{K[2]}e^3 \delta (K[1]-1)dK[1]\right )dK[2] \end{align*}

✓ Sympy. Time used: 0.420 (sec). Leaf size: 58

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

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