Find the Laplace transform of the following function.
f (t) = t^3 H( t − 2) + e^4t H (t − 3)

Question

Find the Laplace transform of the following function.
f (t) = t^3 H( t − 2) + e^4t H (t − 3)

unklerhaukus · Answer

$$f (t) = t^3 H( t − 2) + e^{4t} H (t − 3)$$$$\mathcal L \{f(t)\}=$$

foolaroundmath · Answer

$$\Large L(f(t-\alpha).H(t-\alpha)) = \int_{0}^{\infty}e^{-st}.f(t-\alpha).H(t-\alpha)dt$$$$\Large = \int_{\alpha}^{\infty}e^{-st}f(t-\alpha)dt$$ Taking $z = t-\alpha $ $$\Large = \int_{0}^{\infty}e^{-s(z+\alpha)}f(z)dz = e^{-s\alpha}\int_{0}^{\infty}e^{-sz}f(z)dz $$$$\Large =e^{-s\alpha}L(f(t)) $$So, what you want to do is to express the given function as a sum of f(t-a).H(t-a) and then apply this relation

turingtest · Answer

H is the heavyside function I take it?

unklerhaukus · Answer

i think so

turingtest · Answer

$$\mathcal L\{u_c(t)g(t)\}=e^{-cs}\mathcal L\{g(t+c)\}$$may work

turingtest · Answer

$u_c$ is how I write the heaviside function

unklerhaukus · Answer

my book uses
$$h(t − a) $$

turingtest · Answer

oh for the heaviside, nvmd

turingtest · Answer

same thing$$h(t-a)=h_a(t)$$

unklerhaukus · Answer

yerh for the unit heavyside step function

foolaroundmath · Answer

these set of lectures (specially the second one might help you with this problem) http://home.iitk.ac.in/~sghorai/TEACHING/MTH203/ode17.pdf http://home.iitk.ac.in/~sghorai/TEACHING/MTH203/ode18.pdf http://home.iitk.ac.in/~sghorai/TEACHING/MTH203/ode19.pdf

anonymous · Answer

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