67.22.4 problem 31.6 (d)

Internal problem ID [16918]
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 : 31.6 (d)
Date solved : Thursday, October 02, 2025 at 01:40:21 PM
CAS classification : [[_2nd_order, _quadrature]]

\begin{align*} y^{\prime \prime }&=\delta \left (t -1\right )-\delta \left (-4+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: 0.109 (sec). Leaf size: 23

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

\[ y = \left (-t +4\right ) \operatorname {Heaviside}\left (t -4\right )+\operatorname {Heaviside}\left (t -1\right ) \left (t -1\right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 87

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

\begin{align*} y(t)&\to -t \int _1^0(\delta (K[1]-1)-\delta (K[1]-4))dK[1]+\int _1^t\int _1^{K[2]}(\delta (K[1]-1)-\delta (K[1]-4))dK[1]dK[2]-\int _1^0\int _1^{K[2]}(\delta (K[1]-1)-\delta (K[1]-4))dK[1]dK[2] \end{align*}

✓ Sympy. Time used: 0.392 (sec). Leaf size: 65

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(Dirac(t - 4) - Dirac(t - 1) + 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)
 

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