Any suggestions for finding the inverse Laplace transform of 11/(s^2+16)^2?

In summary, the conversation is about finding the inverse Laplace transform for 11/(s^2+16)^2 and the suggested methods to do so, including using partial fraction decomposition, the integration rule, and the convolution rule.
  • #1
bmed90
99
0
Hi,

I would like to find the inverse Laplace transform for

11/(s^2+16)^2

I have tried to expand it using the following partial fraction decomp to find the constants and take the inverse Laplace but this did not work

C1(s)+ C2/(s^2+16) + C3(s)+C4/(s^2+16)^2

Does anyone have any suggestions?
 
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  • #2
First try to find
$$\mathcal{L}^{-1} \left\{ \frac{1}{(s^2+16)} \right\}$$
Then use the integration rule
$$\mathcal{L}^{-1} \{ \mathrm{F}(s) \} = t \, \mathcal{L}^{-1} \left\{ \int_s^\infty \! \mathrm{F}(u) \, \mathrm{d}u \right\}
\\
\text{or the convolution rule}
\\
\mathcal{L}^{-1} \left\{ G(s)H(s) \right\} = \int_0^t g(t-\tau)h(\tau) \mathop{d\tau}
\\
\text{where}
\\
\mathcal{L}^{-1} \{ \mathrm{G}(s) \} =g(t)
\\
\mathcal{L}^{-1} \{ \mathrm{H}(s) \} =h(t)

$$
 

1. What is an Inverse Laplace transform?

The Inverse Laplace transform is a mathematical operation that takes a function in the complex domain and transforms it into a function in the time domain. It is the inverse operation of the Laplace transform and is used to solve differential equations in engineering and physics.

2. How is an Inverse Laplace transform calculated?

The Inverse Laplace transform can be calculated using various methods such as partial fraction expansion, convolution, and contour integration. The method used depends on the complexity of the function being transformed. There are also tables and software programs that can assist in the calculation.

3. What is the significance of the Inverse Laplace transform?

The Inverse Laplace transform is significant because it allows us to solve differential equations in the time domain, which is more intuitive and easier to interpret than the complex domain. It also has numerous applications in engineering, physics, and other fields.

4. Are there any limitations to the Inverse Laplace transform?

Yes, there are some limitations to the Inverse Laplace transform. It can only be applied to functions that have a Laplace transform, and the function must be causal (it cannot predict the future). Additionally, the function must have a finite number of discontinuities and cannot have an infinite number of poles.

5. How is the Inverse Laplace transform used in real-world applications?

The Inverse Laplace transform has many real-world applications, such as in circuit analysis, control systems, signal processing, and heat transfer. It is also used in the study of fluid dynamics, quantum mechanics, and other fields where differential equations are used to model physical systems.

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