Please i need some help in Partial Fractions

In summary, the common epidemic model assumes that a disease spreads at a rate proportional to the product of the total number of infected and the number of not yet infected individuals. This is represented by the equation \frac {dx}{dt} = k(x+1)(n-x). To solve this problem, one can use the partial fractions method by setting up the equation \int\frac{1}{(x+1)(n-x)}dx = \int kdt. By simplifying the equation, the solution can be obtained as \frac {1}{(x+1)(n-x)} = \frac {1}{n+1} \left( \frac {1}{x+1} + \frac {1}{n
  • #1
ISU20CpreE
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Partial Fractions:
A single infected individual enters a comunnity of n susceptible individuals. Let x be the number of newly infected individuals at time t. The common epidemic model assumes that the disease spreads at a rate proportional to the product of the total number infected and the number not yet infected.So
[tex] \frac {dx} {dt} = k(x+1) (n-x) [/tex] and you obtain [tex] \int\frac {1} {(x+1)(n-x)} dx = \int k dt [/tex] I need to know how to set up the problem and then work from there.

Any suggestions.
 
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  • #2
You have already "set up" the problem.

I think you're looking for this:

[tex]\frac {1}{(x+1)(n-x)} = \frac {1}{n+1} \left( \frac {1}{x+1} + \frac {1}{n-x}\right)[/tex]
 

1. What are partial fractions?

Partial fractions are a method for simplifying rational expressions (fractions) that have polynomials in both the numerator and denominator. It involves breaking down the fraction into smaller, simpler fractions that are easier to work with.

2. When do we use partial fractions?

Partial fractions are commonly used in algebra and calculus when dealing with rational functions. They can also be used in solving differential equations and in integration of trigonometric functions.

3. How do we solve partial fractions?

The steps for solving partial fractions are as follows:
1. Factor the denominator of the given fraction into irreducible polynomials.
2. Write the fraction as a sum of simpler fractions, where the numerator of each fraction is a polynomial with a degree that is one less than the degree of the corresponding irreducible polynomial in the denominator.
3. Equate the coefficients of the like terms on both sides of the equation.
4. Solve for the unknown coefficients by using algebraic manipulation.
5. Combine the fractions into a single expression.

4. Can we always use partial fractions to simplify a rational expression?

No, partial fractions can only be used if the denominator of the rational expression can be factored into distinct linear or quadratic factors. If the denominator cannot be factored, then partial fractions cannot be applied.

5. Are there any real-world applications of partial fractions?

Yes, partial fractions have many real-world applications, such as in electrical engineering for analyzing circuits, in chemistry for solving rate equations, and in statistics for evaluating probabilities.

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