What is the Laplace Transform of #tcosat+sinat#?

Answer 1

# ℒ \ {tcosat+sinat} = (s^2-a^2)/(s^2+a^2)^2 + a/(s^2+a^2) #

Which can also be written as:

# ℒ \ {tcosat+sinat} = (s^2-a^2 + as^2+a^3)/(s^2+a^2)^2 #

We seek:

# ℒ \ {tcosat+sinat} #

First, we observe that the linearity property of the Laplace transformation means that

# ℒ \ {tcosat+sinat} = ℒ \ {tcosat} + ℒ \ {sinat} #

Using a simple Laplace Transform lookup table, we could discover:

# ℒ \ {tcosat} = (s^2-a^2)/(s^2+a^2)^2# # ℒ \ {sinat} = a/(s^2+a^2)#

Thus:

# ℒ \ {tcosat+sinat} = (s^2-a^2)/(s^2+a^2)^2 + a/(s^2+a^2) #
# " " = (s^2-a^2)/(s^2+a^2)^2 + (a(s^2+a^2))/(s^2+a^2)^2 #
# " " = (s^2-a^2 + as^2+a^3)/(s^2+a^2)^2 #

I'm assuming that the above has to be derived, which is simple to do with the help of the derivative's Laplace transformation rule:

# ℒ \ (f''(t))=s^2 F(s)−s f(0)−f'(0) #

Should we allow:

# f(t) = sinat => f(0) = 0#

When we differentiate, we obtain:

# f'(t) = acosat => f'(0)=a# # f''(t) = -a^2sinat # # f''(t) = -a^2f(t) #

Using Laplace transforms, we obtain:

# ℒ \ {f''(t)} = -a^2ℒ \ {f(t)} # # :. s^2 F(s)−s f(0)−f'(0) = -a^2F(s) # # :. s^2 F(s)−0−a = -a^2F(s) # # :. s^2 F(s)+a^2F(s) = a # # :. (s^2 +a^2)F(s) = a # # :. F(s) = a/(s^2 +a^2) #

As a result, we have demonstrated;

# ℒ \ {sinat} = a/(s^2+a^2)# QED

Likewise, let's say that we now

# f(t) = tcosat => f(0) = 0#

Next, using the product rule:

# f'(t) = (t)(-asinat) + (1)(cosat) # # " " = -atsinat+cosat => f'(0) = 1 #

Re-differentiating, we obtain:

# f''(t) = (-at)(acosat) + (-a)(sinat) -asinat# # " " = -a^2tcosat -2asinat # # " " = -a^2f(t) -2asinat => f''(0) = 0#

Using Laplace transforms, we obtain:

# ℒ \ {f''(t)} = -a^2ℒ \ {f(t)} - 2aℒ \ {sin at}# # :. s^2 F(s)−s f(0)−f'(0) = -a^2F(s) - 2a a/(s^2+a^2)#
# :. s^2 F(s)−0−1 = -a^2F(s) - (2a^2)/(s^2+a^2)#
# :. s^2 F(s)+a^2F(s) = 1 - (2a^2)/(s^2+a^2)#
# :. (s^2 +a^2)F(s) = (s^2+a^2- 2a^2)/(s^2+a^2)#
# :. (s^2 +a^2)F(s) = (s^2-a^2)/(s^2+a^2)#
# :. F(s) = (s^2-a^2)/(s^2+a^2)^2#

As a result, we have demonstrated;

# ℒ \ {tcosat} = (s^2-a^2)/(s^2+a^2)^2# QED
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7