Partial fractions ( part of a logistic equation)

In summary: Thus, the partial fraction decomposition for the Bernoulli equation is,$\displaystyle \begin{align*} \frac{1}{K \, N } + \frac{1}{K \left( K - N \right) } \end{align*}$
  • #1
adzix
1
0
Hi everyone, I am stuck on a problem. I need to give a partial fraction of 1/N(k-N). I have tried every method so far ( plotting roots, systems of equations). I think I found A=1/k but I have no clue how to find B value. I would really appreciate any help as I am a desperate student trying to pass the class. (the initial problem is dN/dt=(r/K)N(K-N), N(0)=No)
 
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  • #2
adzix said:
Hi everyone, I am stuck on a problem. I need to give a partial fraction of 1/N(k-N). I have tried every method so far ( plotting roots, systems of equations). I think I found A=1/k but I have no clue how to find B value. I would really appreciate any help as I am a desperate student trying to pass the class. (the initial problem is dN/dt=(r/K)N(K-N), N(0)=No)

$\displaystyle \begin{align*} \frac{A}{N} + \frac{B}{K - N} &\equiv \frac{1}{N \left( K - N \right) } \\ A \left( K - N \right) + B \,N &\equiv 1 \end{align*}$

Let $\displaystyle \begin{align*} N = 0 \end{align*}$ to find $\displaystyle \begin{align*} A\,K = 1 \implies A = \frac{1}{K} \end{align*}$

Let $\displaystyle \begin{align*} N = K \end{align*}$ to find $\displaystyle \begin{align*} B \, K = 1 \implies B = \frac{1}{K} \end{align*}$

so the partial fraction decomposition is $\displaystyle \begin{align*} \frac{1}{K \, N } + \frac{1}{K \left( K - N \right) } \end{align*}$.
 
  • #3
Although not asked for, it might be insightful for the OP to notice that the logistic equation is in fact a special case of the Bernoulli equation, and can be solved without partial fractions! However it would be advisable to ignore if you (the OP) have not yet covered this :).

The Bernoulli equation is of the form:

$\begin{equation}y' = f(x)y^a - g(x)y \end{equation}$

For convenience, let's jot down the logistic equation also,

\(\displaystyle x' = rx - rx^2\).

Quite a resemblance, isn't there? Indeed, the Bernoulli equation with $a = 2$ and $f(x) = g(x) = -r$ is the logistic equation, also known as the Riccati equation.

Returning to the logistic equation and dividing through by $x^2$ gives,

\(\displaystyle x' = rx - rx^2 \implies x^{-2} x' = rx^{-1} - r\).

Now we can make a (perhaps) devious substitution, $u = x^{-1}$.

It follows that, $u' = -x^{-2}x'$.

But we know what $x'$ is, so we can expand this differential equation in $u$,

$u' = -x^{-2}(rx - rx^2) = -r(x^{-1} - 1)$.

Thus, $u' + ru = r ... (1)$

Aha! The differential equation is now in it's linear form. We can use an integrating factor to finish off.

\(\displaystyle I(t) = e^{\int r dt} = e^{rt}.\)

Multiply (1) by $I(t)$,

$e^{rt}u' + e^{rt} ru = e^{rt} r$

We notice the following identity on the left hand side,

$\dfrac{d}{dt}(ue^{rt}) \equiv u'e^{rt} + rue^{rt}$, now we can integrate.$\int \frac{d}{dt}(ue^{rt}) dt = \int re^{rt} dt$

$ue^{rt} = e^{rt} + c$

$u = 1 + ce^{-rt}$

Recall that $u = x^{-1}$,

$x = \dfrac{1}{1 + ce^{-rt}}.$
 

FAQ: Partial fractions ( part of a logistic equation)

What is a partial fraction?

A partial fraction is a mathematical concept used to break down a complex rational expression into simpler fractions. It involves expressing the denominator as a product of linear factors and finding the coefficients for each term in the numerator.

How is partial fraction used in a logistic equation?

In a logistic equation, the partial fraction is used to break down the equation into smaller components. This helps in understanding the behavior of the equation and making predictions about the growth or decay of a population or system.

What is the process for finding partial fractions?

The process for finding partial fractions involves the steps of decomposing the denominator, setting up equations for the unknown coefficients, and solving for the coefficients using algebraic techniques. The final result is a sum of simpler fractions with distinct denominators.

Can partial fractions be used to solve any type of equation?

No, partial fractions can only be used to solve rational expressions with a polynomial in the denominator. It cannot be used for equations with other functions, such as trigonometric or exponential functions.

What are the real-world applications of partial fractions?

Partial fractions have various applications in fields such as engineering, physics, economics, and biology. They are used to model and analyze complex systems, such as population growth, chemical reactions, and electrical circuits.

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