Problem 7: Use the known Laplace Transforms L({e}^( λt)sinωt) = \frac{ ω}{(s - {λ)}^(2) + {ω}^(2) } and L({e}^( λt)cosωt) = \frac{s - λ}{(s - {λ)}^(2) + {ω}^(2) } and the result…

Question

Problem 7: Use the known Laplace Transforms


`L({e}^( λt)sinωt) = \frac{ ω}{(s - {λ)}^(2) + {ω}^(2) }`
and

`L({e}^( λt)cosωt) = \frac{s - λ}{(s - {λ)}^(2) + {ω}^(2) }`
and the result of Exercise 6 to find

`L({te}^( λt)cosωt)`
and

`L({te}^( λt)sinωt)`
​

Answer 1

Let

`F(s) = L\left\{e^(\lambda t) \sin(\omega t)\right\} = (\omega)/((s-\lambda)^2 + \omega^2)`

By the result of ex. 6,

`-F'(s) = L\left\{t e^(\lambda t) \sin(\omega t)\right\} = \boxed{(2\omega (s-\lambda))/(\left((s-\lambda)^2 + \omega^2\right)^2)}`

In a similar way, you'll find that

`L\left\{t e^(\lambda t) \cos(\omega t)\right\} = \boxed{((s-\lambda)^2 - \omega^2)/(\left((s-\lambda)^2 + \omega^2\right)^2)}`