Inverse laplace transform (easy one)

In summary, the conversation discusses finding the inverse Laplace of a given expression using partial fraction decomposition and translation. The final answer is e^3t[3cos(4t) + 5/4sin(4t)], but there is confusion about the coefficient for the sine term. The correct coefficient is 7/2, which is obtained by correctly handling the shift in the bottom term.
  • #1
Derill03
63
0
I am asked to find inverse laplace of (3s+5)/(s^2-6s+25)

I do partial frac and get (3s)/((s-3)^2+16) + (5/(s-3)^2+16)

I do the translation and get [(3s)/s^2+16 + 5/(s^2+16)]*e^3t

My final answer is e^3t[3cos(4t) + 5/4sin(4t)]

but the book says the sin term should be 7/2sin(4t)? I say 5/4 because a 4 needs to be in numerator to make use of the table for k/(s^2+k^2) so if you multiply 5 by (5/4) the 4 can be in numerator as 4/(s^2+16) because it will cancel and be the original 5.

I just need to know how the 7/2 gets in the sin term and why its not (5/4)?
 
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  • #2
You didn't handle the shift correctly. The bottom has (s-3) in it, but the top doesn't for the cosine term. Fixing that will probably fix the sine term.
 

1. What is an inverse Laplace transform?

An inverse Laplace transform is a mathematical operation that takes a function in the Laplace domain and converts it back to its original form in the time domain. This operation is useful in solving differential equations and analyzing the behavior of systems.

2. How is an inverse Laplace transform performed?

An inverse Laplace transform is performed by using a table of Laplace transforms, which contains the relationships between functions in the time domain and their corresponding Laplace transforms. The inverse Laplace transform can also be computed using integration techniques.

3. What is the difference between a Laplace transform and an inverse Laplace transform?

A Laplace transform converts a function from the time domain to the Laplace domain, while an inverse Laplace transform converts a function from the Laplace domain back to the time domain. They are inverse operations of each other and are used in conjunction to solve differential equations.

4. What are some applications of inverse Laplace transforms?

Inverse Laplace transforms have various applications in engineering, physics, and other fields. They are used to solve differential equations in control systems, to analyze the stability of systems, and to model the behavior of electric circuits, among others.

5. Can all functions be inverted using the Laplace transform?

No, not all functions can be inverted using the Laplace transform. Some functions may not have a corresponding Laplace transform or may have a complex-valued Laplace transform, which cannot be inverted using simple integration techniques. In these cases, other methods such as numerical methods or partial fraction decomposition may be used.

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