Question

Use the fact that L{f'}=sF-f(0) to derive the formula for L{cosh at}

Answer 1

cosh(at) = d/dt (1/a sinh(at))

sinh(at) = d/dt (1/a cosh(at))
So,

Since L{f(at)} = 1/|a| F(s/a)
L{cosh(at)} = sL{1/a sinh(t)}
= s/a L{sinh(t)}

L{sinh(at)} = sL{1/a cosh(at)}
= s/a L{cosh(at)} - s/a

So,

L{cosh(at)} = s/a (s/a L{cosh(at)} - s/a
L{cosh(at)}(1-s^2/a^2) = -s/a
= (s/a^2)/(s^2/a^2 - 1)
= s/(s^2-a^2)

Better check my algebra in the middle steps. I think I dropped an "a" somewhere, but the final answer is correct. You work this in the same way you do integration by parts twice for trig functions.