If y(a) is a unique function which is continuous on [0,
You can select a piecewise continuous function, if all other possible functions, y (a) are discontinuous, to be the inverse transform.
L-1[Y(b)](a)
Definition of Inverse Laplace Transform
An integral defines the laplace transform Y(b) of a function y(a) defined on [o,
Also, the formula to determine y(a) if Y(b) is given, involves an integral.
Inverse Laplace transform table
To compute the inverse transform, we will use the table:
|
Function y(a) |
Transform Y(b) |
b |
|
1 |
\(\begin{array}{l}\frac{1}{b}\end{array} \)
|
b>0 |
|
a |
\(\begin{array}{l}\frac{1}{b^2}\end{array} \)
|
b>0 |
|
Ai , i = integer |
\(\begin{array}{l}\frac{i!}{s^(i+1)}\end{array} \)
|
b>0 |
|
exp (ta), where t = constant |
\(\begin{array}{l}\frac{1}{(b-t)}\end{array} \)
|
b>t |
|
cos (sa), s= constant |
\(\begin{array}{l}\frac{b}{b^2+s^2}\end{array} \)
|
b>0 |
|
Sin (sa), s = constant |
\(\begin{array}{l}\frac{t}{b^2+s^2}\end{array} \)
|
b>0 |
|
exp(ta)cos(sa) |
\(\begin{array}{l}\frac{b-t}{(b-t)^2 +s^2}\end{array} \)
|
b>t |
|
exp(ta)sin(sa) |
\(\begin{array}{l}\frac{s}{(b-t)^2 +s^2}\end{array} \)
|
b>t |
Example of Inverse Laplace
Example: Find the inverse transform of each of the following.
Y(b)=
Solution:
Step 1: The first term is a constant as we can see from the denominator of the first term.
Step 2: Before taking the inverse transform, let’s take the factor 6 out, so the correct numerator is 6.
Step 3: The second term has an exponential t = 8.
Step 4: The numerator is perfect.
Step 5: The third term is also an exponential, t= 3.
Step 6: Now before taking the inverse transforms, we need to factor out 4 first.
Y(b) = 6
y(a) = 6(1) – e8a +4 (e3t)
= 6 – e8a +4 e3t
Inverse Laplace Transform Calculator
The Inverse Laplace Transform Calculator helps in finding the Inverse Laplace Transform Calculator of the given function.
Remember, L-1[Y(b)](a) is a function that y(a) that L(y(a) )= Y(b). So, generally, we use this property of linearity of Laplace transform to find the Inverse Laplace transform.
For the Laplace Transform, you can also use the Laplace Calculator.
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