Laplace Transform of t cost | L{t cos(at)}

The function t cos(t) is the product of t and cosine of t. The Laplace transform of tcos(t) is (s²-1)/(s²+1)². In this article, we will find the Laplace transform of both tcos(t) and tcos(at).

What is the Laplace Transform of t cos(t)?

Answer: The Laplace transform of t cos t is (s²-1)/(s²+1)².

Proof:

We know that the Laplace transform of a function f(t) multiplied by t, denoted by L{t f(t)}, is given by the following multiplication by t Laplace transform formula:

$L\{t f(t)\} = – \dfrac{d}{ds}(F(s))$, where L{f(t)}=F(s) …(∗)

Step 1: Put f(t) = cos(t) in the above formula.

∴ F(s) = L{f(t)} = L{cos(t)} = s/(s²+1)

Step 2: Now, by the formula (∗), the Laplace transform of tcos(t) is equal to

$L\{t\cos(t)\} = – \dfrac{d}{ds}\left(\dfrac{s}{s^2+1}\right)$

Step 3: Applying the quotient rule of derivatives, we obtain that

$L\{t\cos(t)\}$ $= – \dfrac{(s^2+1)\frac{d}{ds}(s)-s \frac{d}{ds}(s^2+1)}{(s^2+1)^2}$

$= – \dfrac{(s^2+1)\cdot 1-s \cdot 2s}{(s^2+1)^2}$

$= – \dfrac{s^2+1-2s^2}{(s^2+1)^2}$

$= \dfrac{s^2-1}{(s^2+1)^2}$.

So the Laplace transform of tcos t is (s²-1)/(s²+1)².

Find the Laplace transform of t cos(t). Summary: L{t cos t} = (s2-1)/(s2+1)2.

Also Read:

Laplace transform of t:	1/s2
Laplace transform of sin t:	1/(s2+1)
Laplace transform of sin(t)/t:	tan-1(1/s)
Laplace transform of cos t:	s/(s2+1)
Laplace transform of cos(t)/t:	Does Not Exist
Laplace transform of e-t:	1/(s+1)
Laplace transform of 1:	1/s

What is the Laplace Transform of t cos(at)?

Answer: The Laplace transform of t cos at is (s²-a²)/(s²+a²)².

Proof:

In the above formula (∗), we put f(t) = t cos(at). As L{cos at} = s/(s²+a²), the Laplace transform of t cos(at) by the above formula (∗) will be equal to

$L\{t\cos(at)\} = – \dfrac{d}{ds}\left(\dfrac{s}{s^2+a^2}\right)$

$= – \dfrac{(s^2+a^2)\frac{d}{ds}(s)-s \frac{d}{ds}(s^2+a^2)}{(s^2+a^2)^2}$

$= – \dfrac{(s^2+a^2)\cdot 1-s \cdot 2s}{(s^2+a^2)^2}$

$= – \dfrac{s^2+a^2-2s^2}{(s^2+a^2)^2}$

$= \dfrac{s^2-a^2}{(s^2+a^2)^2}$.

So the Laplace transform of tcos at is (s²-a²)/(s²+a²)².

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