MATH 20D Lecture 22

Example. Let $f(t)$ be a function with Laplace transform $\mathcal{L}\{f(t)\}=F(s)$. Find $\mathcal{L}\{tf(t)\}$ in terms of $F(s)$.

So

Example. Let $f(t)$ be a function with Laplace transform $\mathcal{L}\{f(t)\}=F(s)$. Find $\mathcal{L}\{e^{at}f(t)\}$ in terms of $F(s)$ for any fixed $a$.

Recall,
Replace by in this formula

Thus, where
“Shift the transform of by a unit” (i.e. replace in by )

Examples

1. for .
   for or
2. 
   Here for
   for or
3. 
   First for
   so for or

Example. Let $f(t)$ be a function with Laplace transform $\mathcal{L}\{f(t)\}$. Find $\mathcal{L}\{f^{\prime }(t)\}$ and $\mathcal{L}\{f^{\prime \prime }(t)\}$

When , the limit D.N.E but when , this limit
Thus, when ,

Therefore, when ,

deriv. of


for

for

for

Summary. Our results so far

If then

• and