67.4.30 problem Problem 5(a)

Internal problem ID [14074]
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 : Tuesday, January 28, 2025 at 06:13:40 AM
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*}

✓ Solution by Maple

Time used: 11.590 (sec). Leaf size: 44

dsolve([diff(y(t),t$2)+(2*Pi)^2*y(t)=3*Dirac(t-1/3)-Dirac(t-1),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.120 (sec). Leaf size: 172

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

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