text{Laplace transforms A powerful tool in solving problems inengineering and physics is the Laplace transform.

Question

$Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by$
$(F (s) = \int_{0}^{\infty} e^{- s t} f (t) d t$
$where we assume s is a positive real number. For example, to find the Laplace transform of$ $f (t) = e^{- t}, the following improper integral is evaluated using integration by parts:$
$F (s) = \int_{0}^{\infty} e^{- s t} e^{- t} d t = \int_{0}^{\infty} e^{- (s + 1) t} d t = \frac{1}{s + 1}$
$Verify the following Laplace transforms, where u is a real number.$
$f (t) = t \to F (s) = \frac{1}{s^{2}}$

Answer 1

$Step 1$
$Here, the objective is to prove that L (t) = \frac{1}{s^{2}}$
$Step 2$
$Let f (t) be t$
$Laplace transformation of f(t) is defined as:$
$L (t) = F (s) = \int_{0}^{\infty} e^{- s t} f (t) d t$
$Substitute f (t) = t$
$L (t) = F (s) = \int_{0}^{\infty} e^{- s t} t d t$
\(\text{Use integration by part } \int fg=fg-int

text{Laplace transforms A powerful tool in solving problems inengineering and physics is the Laplace transform.

Want to know more about Laplace transform?

Expert Answer

Not exactly what you’re looking for?

Relevant Questions