determine the inverse laplace trasnform of a function below 7s+56/s^2+125+85

To find the inverse Laplace transform of the function \( \frac{7s + 56}{s^2 + 125 + 85} \), we first need to simplify the given expression.

The function is:
\[ \frac{7s + 56}{s^2 + 125 + 85} \]

First, let's rewrite the denominator:
\[ s^2 + 125 + 85 = s^2 + 210 \]

So the function becomes:
\[ \frac{7s + 56}{s^2 + 210} \]

Now, we decompose the function into simpler parts. Let's write it as:
\[ \frac{7s}{s^2 + 210} + \frac{56}{s^2 + 210} \]

We recognize that the first part \( \frac{7s}{s^2 + 210} \) corresponds to the inverse Laplace transform involving a cosine function, and the second part \( \frac{56}{s^2 + 210} \) corresponds to a sine function.

1. For \( \frac{7s}{s^2 + 210} \):

This is of the form \( \frac{s}{s^2 + a^2} \), whose inverse Laplace transform is \( \cos(\sqrt{a} t) \).

Here, \( a^2 = 210 \), so \( a = \sqrt{210} \).

Therefore, the inverse Laplace transform of \( \frac{7s}{s^2 + 210} \) is:
\[ 7 \cos(\sqrt{210} t) \]

2. For \( \frac{56}{s^2 + 210} \):

This is of the form \( \frac{a}{s^2 + a^2} \), whose inverse Laplace transform is \( \frac{\sin(\sqrt{a} t)}{\sqrt{a}} \).

Here, \( a^2 = 210 \), so \( a = \sqrt{210} \).

Therefore, the inverse Laplace transform of \( \frac{56}{s^2 + 210} \) is:
\[ \frac{56}{\sqrt{210}} \sin(\sqrt{210} t) \]

Combining both parts, the inverse Laplace transform of the given function is:
\[ \mathcal{L}^{-1} \left\{ \frac{7s + 56}{s^2 + 210} \right\} = 7 \cos(\sqrt{210} t) + \frac{56}{\sqrt{210}} \sin(\sqrt{210} t) \]

This is the final result.

Would you like more details or have any questions?

Here are some related questions you might be interested in:
1. How do you determine the inverse Laplace transform of a more complex function?
2. What are the standard forms of Laplace transforms and their inverses?
3. How do you use partial fraction decomposition in Laplace transforms?
4. What is the significance of the \( \cos \) and \( \sin \) terms in the inverse Laplace transform?
5. Can you explain the properties of the Laplace transform?
6. How can the Laplace transform be used to solve differential equations?
7. What are some common applications of the Laplace transform in engineering and physics?
8. How do you handle inverse Laplace transforms involving higher-order polynomials in the denominator?

**Tip:** When dealing with Laplace transforms, always check if the function can be simplified or decomposed into known transform pairs to make finding the inverse easier.

Discover how to find the inverse Laplace transform of (7s + 56) / (s^2 + 210) step-by-step. This problem involves applying the inverse Laplace transform formula and using partial fraction decomposition to simplify the expression. Suitable for college-level students and enthusiasts of advanced mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation