How should I inverse Laplace transform this function? ((omega_n)^2)/(s^2+bs+c)

limfne2c 2022-10-17 Answered
How should I inverse Laplace transform this function?
( ω n ) 2 s 2 + b s + c
You can still ask an expert for help

Want to know more about Laplace transform?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

espava8b
Answered 2022-10-18 Author has 12 answers
In such a case the most convenient way to compute the inverse transform is to use standard results from Laplace transform tables. The transform pairs that you need here can be found in most tables:
1 s 2 t u ( t ) ω 0 s 2 + ω 0 2 sin ( ω 0 t ) u ( t ) ω 0 s 2 ω 0 2 sinh ( ω 0 t ) u ( t ) f ( t ) F ( s ) e α t f ( t ) F ( s + α )
Now rewrite the function as
X ( s ) = ω n 2 s 2 + b s + c = ω n 2 ( s + b / 2 ) 2 + c b 2 / 4
You have to distinguish 3 cases:
c > b 2 / 4: Define ω 0 2 = c b 2 / 4
X ( s ) = ω n 2 ω 0 ω 0 ( s + b / 2 ) 2 + ω 0 2 ω n 2 ω 0 e b t / 2 sin ( ω 0 t ) u ( t )
c < b 2 / 4: Define ω 0 2 = b 2 / 4 c
X ( s ) = ω n 2 ω 0 ω 0 ( s + b / 2 ) 2 ω 0 2 ω n 2 ω 0 e b t / 2 sinh ( ω 0 t ) u ( t )
c = b 2 / 4
X ( s ) = ω n 2 1 ( s + b / 2 ) 2 ω n 2 e b t / 2 t u ( t )
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2021-02-09
Solve the initial value problem below using the method of Laplace transforms
2ty3ty+3y=6,y(0)=2,y(0)=3
asked 2021-12-30
Consider a bacterial population that grows according to the function f(t)=500e0.05t measured in minutes. After 4 hours, how many germs are present in the population? When will the population of bacteria reach 100 million?
asked 2021-12-27
You drink a Starbucks Coffee in SM Seaside with a 120 mg. of caffeine. Each hour, the caffeine in your body system reduces by about 12 percent. How long will you have a 10 mg. of caffeine?
asked 2022-01-15
I was toying with equations of the type f(x+α)=f(x) where f is a real function. For example if α=π2 then the solutions include the function fλ,μ(x)=λcos(x+μ). Are there more solutions?
On the other hand, if I want to solve the equation for any α, I can assume a solution of the form f(x)=eλx, and find λ as a complex number that enables me to solve the equation...
I was wondering: is the set of the solutions of dimension 2 because the derivative operator creates one dimension and the operator ϕ:f(x)f(x+1) adds another one? Is there some litterature about this kind of equations?
asked 2022-11-24
How to find the Direct Discrete Laplace Transform of ( 2 n n )
asked 2021-12-29
Find a logistic function that describes the given population. Then graph the population function. The population increases from 300 to 700 in the first year and eventually levels off at 5400. Write the equation of a logistic function that models the given population. P(t)= (Type an exact answer.)
asked 2022-09-14
How the answer was computed for the following infinite series:
x = 1 e s x p q x 1 = p e s 1 q e s

New questions