71.16.6 problem 6

Internal problem ID [14555]
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 : 6
Date solved : Tuesday, January 28, 2025 at 06:43:24 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\cos \left (x \right ) \delta \left (x -\pi \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 9.609 (sec). Leaf size: 18

dsolve([diff(y(x),x$2)+4*y(x)=cos(x)*Dirac(x-Pi),y(0) = 0, D(y)(0) = 1],y(x), singsol=all)
 

\[ y = -\frac {\sin \left (2 x \right ) \left (-1+\operatorname {Heaviside}\left (x -\pi \right )\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.031 (sec). Leaf size: 62

DSolve[{D[y[x],{x,2}]+4*y[x]==Cos[x]*DiracDelta[x-Pi],{y[0]==0,Derivative[1][y][0] ==1}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{2} \left (-2 \sin (2 x) \int _1^0-\frac {1}{2} \delta (\pi -K[1])dK[1]+2 \sin (2 x) \int _1^x-\frac {1}{2} \delta (\pi -K[1])dK[1]+\sin (2 x)\right ) \]