Find the inverse Laplace transform of $\frac{s+1}{s(s-4)}$
Can anybody help me, please.
Find the inverse Laplace transform of $\frac{s+1}{s(s-4)}$
Can anybody help me, please.
HINT: Use partial fraction decomposition, i.e. determine values of $a$ and $b$ that make the following into an inequality: $ \frac{s+1}{s(s-4)} = \frac{a}{s} + \frac{b}{s-4} $ Now use table of Laplace transforms.
$\large {\bf Hint}$ Using partial fraction decomposition:
$\frac 1 {s(s-4)}=\frac 1 4\left(\frac{1}{s-4}-\frac 1 {s}\right)$
So we end up with
$\frac 1 {s(s-4)}=\frac 1 4\left(\frac{s+1}{s-4}-\frac {s+1} {s}\right)$
$\frac 1 {s(s-4)}=\frac 1 4\left(1+\frac{5}{s-4}-1-\frac 1 s\right)$
$\frac 1 {s(s-4)}=\frac 1 4\left(\frac{5}{s-4}-\frac 1 s\right)$
Now recall that the Lapalce Transform is linear, that $\mathscr{L}\{1\}(s)=\frac{1}{s}$ and the shift theorem:
If $\mathscr{L}\{f(t)\}=F(s)$
then $\mathscr{L}\{e^{at}f(t)\}=F(s-a)$
Since this is the third time you have asked about inverse Laplace transforming I would like to offer you a general solution strategy for such problems.
Step 1: Memorize (or at least learn to recognize) Laplace Transforms (LT) of "standard" functions like polynomials, trigonometric functions, exponential function, etc. Keep a list of standard LTs handy.
Step 2: Get comfortable with the different properties of LTs. You should be able to answer questions such as: What happens to the LT when a function is integrated? when it is multiplied by an exponential? when you take a derivative? when it is multiplied by a polynomial?
Step 3: Now look at your problem. Ask yourself: Can I express this function in terms of some other functions whose LTs I have memorized? Note that here you are combining your knowledge from both Steps 1 and 2.
See if you can apply this strategy to the solutions you have seen so far. Why did Sasha and Peter go for a partial fraction decomposition? Because it gave them known functions that are in the standard table of LTs (Step 1). Why did they go for the exp-shift property? Because they saw something like 1/(s-a) instead of a plain 1/s and it immediately shifted their attention to the table of properties of LT (Step 2). As a result they were able to express their problem in terms of functions whose LTs are known (Step 3).