71.16.3 problem 3

Internal problem ID [14552]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 5. The Laplace Transform Method. Exercises 5.5, page 273
Problem number : 3
Date solved : Tuesday, January 28, 2025 at 06:43:21 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+9 y&=\delta \left (x -\pi \right )+\delta \left (x -3 \pi \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: 9.522 (sec). Leaf size: 23

dsolve([diff(y(x),x$2)+9*y(x)=Dirac(x-Pi)+Dirac(x-3*Pi),y(0) = 0, D(y)(0) = 0],y(x), singsol=all)
 

\[ y = -\frac {\left (\operatorname {Heaviside}\left (x -3 \pi \right )+\operatorname {Heaviside}\left (x -\pi \right )\right ) \sin \left (3 x \right )}{3} \]

✓ Solution by Mathematica

Time used: 0.060 (sec). Leaf size: 79

DSolve[{D[y[x],{x,2}]+9*y[x]==DiracDelta[x-Pi]+DiracDelta[x-3*Pi],{y[0]==0,Derivative[1][y][0] ==0}},y[x],x,IncludeSingularSolutions -> True]
 

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