Internal problem ID [12500]
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: 7.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+y a^{2}=\left (\delta \left (x -\pi \right )\right ) f \left (x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.079 (sec). Leaf size: 24
dsolve([diff(y(x),x$2)+a^2*y(x)=Dirac(x-Pi)*f(x),y(0) = 0, D(y)(0) = 0],y(x), singsol=all)
\[ y \left (x \right ) = \frac {\operatorname {Heaviside}\left (x -\pi \right ) f \left (\pi \right ) \sin \left (a \left (x -\pi \right )\right )}{a} \]
✓ Solution by Mathematica
Time used: 0.398 (sec). Leaf size: 26
DSolve[{y''[x]+a^2*y[x]==DiracDelta[x-Pi]*f[x],{y[0]==0,y'[0]==0}},y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {f(\pi ) \theta (x-\pi ) \sin (a (\pi -x))}{a} \]