Problem 02 | Linearity Property of Laplace Transform

Problem 02
By using the linearity property, show that

L(coshat)=ss2a2

 

Solution 02
f(t)=coshat

L{f(t)}=0estf(t)dt

L(coshat)=0estcoshatdt
 

But
coshat=eat+eat2
 

Thus,
L(coshat)=0est(eat+eat2)dt

Linearity Property | Laplace Transform

Linearity Property
If   a   and   b   are constants while   f(t)   and   g(t)   are functions of   t   whose Laplace transform exists, then
 

L{af(t)+bg(t)}=aL{f(t)}+bL{g(t)}

 

Proof of Linearity Property
L{af(t)+bg(t)}=0est[af(t)+bg(t)]dt