Collin's questions via email about Inverse Laplace Transforms

In summary, the first conversation discusses how to evaluate a Laplace inverse of a function with a shifted denominator. The second conversation demonstrates how to use the rule for integrating logarithms to evaluate a Laplace inverse of a logarithmic function. The final result is a function with a shifted denominator.
  • #1
Prove It
Gold Member
MHB
1,465
24
Evaluate $\displaystyle \begin{align*} \mathcal{L}^{-1}\,\left\{ \frac{5\,s^2 + 20\,s + 26}{\left( s + 3 \right) ^3} \right\} \end{align*}$

As the denominator is a function of s + 3, it suggests a shift had to have been utilised. As such, we also need the numerator to be a function of s + 3...

Let $\displaystyle \begin{align*} u = s + 3 \end{align*}$, then $\displaystyle \begin{align*} s = u-3 \end{align*}$ and thus

$\displaystyle \begin{align*} 5\,s^2 + 20\,s + 26 &= 5\,\left( u - 3 \right) ^2 + 20\,\left( u - 3 \right) + 26 \\ &= 5\,\left( u^2 - 6\,u + 9 \right) + 20\,u - 60 + 26 \\ &= 5\,u^2 - 30\,u + 45 + 20\,u - 34 \\ &= 5\,u^2 - 10\,u + 11 \\ &= 5\,\left( s + 3 \right) ^2 - 10\,\left( s + 3 \right) + 11 \end{align*}$

and thus

$\displaystyle \begin{align*} \mathcal{L}^{-1}\,\left\{ \frac{5\,s^2 + 20\,s + 26 }{\left( s + 3 \right) ^3} \right\} &= \mathcal{L}^{-1}\,\left\{ \frac{5\,\left( s + 3 \right) ^2 - 10\,\left( s + 3 \right) + 11}{\left( s + 3 \right) ^3} \right\} \\ &= \mathrm{e}^{-3\,t}\,\mathcal{L}^{-1}\,\left\{ \frac{5\,s^2 - 10\,s + 11}{s^3} \right\} \\ &= \mathrm{e}^{-3\,t}\,\mathcal{L}^{-1}\,\left\{ \frac{5}{s} - \frac{10}{s^2} + \frac{11}{s^3} \right\} \\ &= \mathrm{e}^{-3\,t}\,\left( 5\,\mathcal{L}^{-1}\,\left\{ \frac{0!}{s^{0 + 1}} \right\} - 10\,\mathcal{L}^{-1}\,\left\{ \frac{1!}{s^{1 + 1}} \right\} + \frac{11}{2}\,\mathcal{L}^{-1}\,\left\{ \frac{2!}{s^{2 + 1}} \right\} \right) \\ &= \mathrm{e}^{-3\,t}\,\left( 5\,t^0 - 10\,t^1 + \frac{11}{2}\,t^2 \right) \\ &= \mathrm{e}^{-3\,t} \,\left( 5 - 10\,t + \frac{11}{2}\,t^2 \right) \end{align*}$
Evaluate $\displaystyle \begin{align*} \mathcal{L}^{-1}\,\left\{ 7\log{ \left( \frac{7 + s}{s} \right) } \right\} \end{align*}$

As logarithms have a very simple integral - becoming a rational function, I would make use of this rule:

$\displaystyle \begin{align*} \mathcal{L}\,\left\{ \frac{f\left( t \right) }{t} \right\} = \int_s^{\infty}{ F\left( u \right) \,\mathrm{d}u } \end{align*}$

Now we should note that

$\displaystyle \begin{align*} \int{\left( \frac{1}{7 + u} - \frac{1}{u} \right) \,\mathrm{d}u } &= \log{ \left| 7 + u \right| } - \log{ \left| u \right| } + C \\ &= \log{ \left| \frac{7 + u}{u} \right| } + C \end{align*}$

and thus (since $\displaystyle \begin{align*} s > 0 \end{align*}$)

$\displaystyle \begin{align*} \int_s^{\infty}{ \left( \frac{1}{7 + u} - \frac{1}{u} \right) \,\mathrm{d}u } &= \lim_{b \to \infty} \int_s^b{ \left( \frac{1}{7 + u} - \frac{1}{u} \right) \,\mathrm{d}u } \\ &= \lim_{b \to \infty} \left[ \log{ \left( \frac{7 + u}{u} \right) } \right] _s^b \\ &= \lim_{b \to \infty} \left[ \log{\left( \frac{7 + b}{b} \right) } \right] - \log{ \left( \frac{7 + s}{s} \right) } \\ &= \lim_{b \to \infty} \left[ \log{ \left( \frac{7}{b} + 1 \right) } \right] - \log{ \left( \frac{7 + s}{s} \right) } \\ &= \log{ \left( 0 + 1 \right) } - \log{ \left( \frac{7 + s}{s} \right) } \\ &= 0 -\log{ \left( \frac{7 + s}{s} \right) } \\ &= -\log{ \left( \frac{7 + s}{s} \right) } \end{align*}$

So this suggests that $\displaystyle \begin{align*} F\left( s \right) = - \left( \frac{1}{7 + s} - \frac{1}{s} \right) = \frac{1}{s} - \frac{1}{7 + s} \end{align*}$, therefore

$\displaystyle \begin{align*} f\left( t \right) &= \mathcal{L}^{-1}\,\left\{ \frac{1}{s} - \frac{1}{7 + s} \right\} \\ &= \mathcal{L}^{-1}\,\left\{ \frac{0!}{s^{0 + 1}} \right\} - \mathrm{e}^{-7\,t}\,\mathcal{ L }^{-1}\,\left\{ \frac{0!}{s^{0 + 1}} \right\} \\ &= t^0 - \mathrm{e}^{-7\,t}\,t^0 \\ &= 1 - \mathrm{e}^{-7\,t} \end{align*}$

so finally $\displaystyle \begin{align*} \mathcal{L}^{-1}\,\left\{ 7 \log{ \left( \frac{7 + s}{s} \right) } \right\} &= 7\,\left( \frac{1 - \mathrm{e}^{-7\,t}}{t} \right) \end{align*}$.
 
  • Like
Likes benorin
Mathematics news on Phys.org
  • #2
This is correct. Ty been wanting to study integral transforms recently.
 

FAQ: Collin's questions via email about Inverse Laplace Transforms

1. What are Inverse Laplace Transforms?

Inverse Laplace Transforms are mathematical operations that allow us to convert a function or equation from the Laplace domain back to the time domain. They are the inverse of the Laplace transform, which is a mathematical tool used to simplify differential equations.

2. Why are Inverse Laplace Transforms important?

Inverse Laplace Transforms are important because they allow us to solve complex differential equations in the time domain. This is useful in many fields such as engineering, physics, and mathematics. It also allows us to analyze the behavior of systems over time.

3. How do I perform an Inverse Laplace Transform?

To perform an Inverse Laplace Transform, you need to use a table of Laplace transforms or a computer software program. You will need to know the function or equation in the Laplace domain and use the appropriate formula or technique to convert it back to the time domain.

4. What are some common techniques for finding Inverse Laplace Transforms?

Some common techniques for finding Inverse Laplace Transforms include using partial fraction decomposition, using the convolution theorem, and using the Laplace transform table. Each technique has its own advantages and is suitable for different types of functions or equations.

5. What are some applications of Inverse Laplace Transforms?

Inverse Laplace Transforms have many applications in fields such as control systems, signal processing, and circuit analysis. They are also used in solving differential equations that arise in physics and engineering problems. Additionally, they can be used to model and analyze the behavior of complex systems over time.

Similar threads

Back
Top