67.4.30 problem Problem 5(a)

Internal problem ID [13995]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 5(a)
Date solved : Wednesday, March 05, 2025 at 10:25:05 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 \pi ^{2} y&=3 \delta \left (t -\frac {1}{3}\right )-\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: 11.590 (sec). Leaf size: 44

ode:=diff(diff(y(t),t),t)+4*Pi^2*y(t) = 3*Dirac(t-1/3)-Dirac(t-1); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {\left (-3 \sqrt {3}\, \cos \left (2 \pi t \right )-3 \sin \left (2 \pi t \right )\right ) \operatorname {Heaviside}\left (t -\frac {1}{3}\right )-2 \sin \left (2 \pi t \right ) \operatorname {Heaviside}\left (t -1\right )}{4 \pi } \]

✓ Mathematica. Time used: 0.12 (sec). Leaf size: 172

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

\[ y(t)\to -\sin (2 \pi t) \int _1^0-\frac {\cos (2 \pi K[2]) (\delta (K[2]-1)-9 \delta (3 K[2]-1))}{2 \pi }dK[2]+\sin (2 \pi t) \int _1^t-\frac {\cos (2 \pi K[2]) (\delta (K[2]-1)-9 \delta (3 K[2]-1))}{2 \pi }dK[2]-\cos (2 \pi t) \int _1^0\frac {(\delta (K[1]-1)-9 \delta (3 K[1]-1)) \sin (2 \pi K[1])}{2 \pi }dK[1]+\cos (2 \pi t) \int _1^t\frac {(\delta (K[1]-1)-9 \delta (3 K[1]-1)) \sin (2 \pi K[1])}{2 \pi }dK[1] \]

✓ Sympy. Time used: 1.989 (sec). Leaf size: 139

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(Dirac(t - 1) - 3*Dirac(t - 1/3) + 4*pi**2*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)
 

\[ y{\left (t \right )} = \left (\frac {- \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} \sin {\left (2 \pi t \right )}\, dt - \int \limits ^{0} \left (- 3 \operatorname {Dirac}{\left (t - \frac {1}{3} \right )} \sin {\left (2 \pi t \right )}\right )\, dt}{2 \pi } + \frac {\int \left (\operatorname {Dirac}{\left (t - 1 \right )} - 3 \operatorname {Dirac}{\left (t - \frac {1}{3} \right )}\right ) \sin {\left (2 \pi t \right )}\, dt}{2 \pi }\right ) \cos {\left (2 \pi t \right )} + \left (\frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} \cos {\left (2 \pi t \right )}\, dt + \int \limits ^{0} \left (- 3 \operatorname {Dirac}{\left (t - \frac {1}{3} \right )} \cos {\left (2 \pi t \right )}\right )\, dt}{2 \pi } - \frac {\int \left (\operatorname {Dirac}{\left (t - 1 \right )} - 3 \operatorname {Dirac}{\left (t - \frac {1}{3} \right )}\right ) \cos {\left (2 \pi t \right )}\, dt}{2 \pi }\right ) \sin {\left (2 \pi t \right )} \]