Laplace X-forms Proofs

smms25

Hey, I have been studying differential equations a bit and was wanting some help on some proofs. There are 3 laplace transforms I would like proofs for. Not really sure where to get started or if someone could lead me to place that has these proofs I would greatly appreciate it. Thank you.

Attached Thumbnails

 

HallsofIvy

Have you tried just integrating?

The definition of the Laplace transform of f(x) is
[tex]\int_0^\infty e^{-sx}f(x)dx[/tex]

The transform of the delta function is pretty close to trivial:
[tex]\int_0^\infty e^{-sx}\delta(x)dx= e^{0x}= 1[/tex]

For [itex]f(x)= x^ne^{ax}u(x)[/itex] do repeated integration by parts letting [itex]u= x^n[/itex], [itex]dv= e^{ax}u(x)[/itex] until you have reduced the power of x to 0.

For [itex]f(t-t_0)u(t-t_0)[/itex] do the obvious substitution: let [itex]v= t- t_0[/itex].