Finding a fundamental solution of a linear differential operator $f''-c^2f$, I try in 12.3.1 on Windows 10
DSolve[f''[x] - c^2*f[x] == DiracDelta[x - y], f[x], x]
{{f[x] -> E^(c x) C[1] + E^(-c x) C[2] - ( E^(-c x - c y) (-E^(2 c x) + E^(2 c y)) HeavisideTheta[x - y])/( 2 c)}}
To be sure, I also execute (following the documentation to DiracDelta
, namely the "Properties" section,
and Encyclopedia of Mathematics)
DSolve[f''[x] - c^2*f[x] == DiracDelta[-x + y], f[x], x]
{{f[x] -> E^(c x) C[1] + E^(-c x) C[2] + E^(-c x) (E^(2 c x) Inactive[Integrate][(E^(-c K[1]) DiracDelta[y - K[1]])/( 2 c), {K[1], 1, x}] + Inactive[Integrate][-((E^(c K[2]) DiracDelta[y - K[2]])/( 2 c)), {K[2], 1, x}])}}
Which of these different results is correct?
DiracDela[x]== DiracDelta[-x] ->True
butDiracDela[x-y]== DiracDelta[y-x] ->unevaluated
$\endgroup$DSolve[f''[x] - c^2*f[x] == eps/Pi/(eps^2 + (x - y)^2), f[x], x]
and theneps
tends to zero from above. The result isE^(c x) C[1] + E^(-c x) C[2]
. $\endgroup$eps->0
is not alowed! $\endgroup$