Question

laplace transform example

Answer 1

EXAMPLE OF LAPLACE TRANSFORM

Let f(t)=sin(at), then ℒ{f(t)}=∫[0^-,∞]e^-stsin(at)dt, by definition of Laplace function.

Let u=f(t)=sin(at), du=f'(t)dt=acos(at)dt,

dv=e^-stdt, v=-(1/s)e^-st.

Integrating by parts:

ℒ{f(t)}=-e^-stf(t)/s+(1/s)∫e^-stf'(t)dt=ℒ{sin(at)}=-e^-stsin(at)/s+(a/s)∫e^-stcos(at)dt for t∈[0^-,∞],

Therefore, ℒ{sin(at)}=-e^-stsin(at)/s+(a/s)ℒ{cos(at)}=(a/s)ℒ{cos(at)}, because the first term is zero when the limits are applied.

Now we have to evaluate ℒ{cos(at)}, so let u=cos(at), du=-asin(at). 

ℒ{cos(at)}=-e^-stcos(at)/s-(a/s)∫e^-stsin(at)dt for t∈[0^-,∞]=(1/s)-(a/s)ℒ{sin(at)}, when limits are applied.

ℒ{sin(at)}=(a/s)ℒ{cos(at)}=(a/s)[(1/s)-(a/s)ℒ{sin(at)}].

ℒ{sin(at)}+(a²/s²)ℒ{sin(at)}=a/s²,

ℒ{sin(at)}(1+(a²/s²))=a/s², 

ℒ{sin(at)}=a/(s²+a²), which is the Laplace Transform of sin(at).