Question

Find the partial decomposition of f(x)?

Find the partial decomposition of f(x) in the following logistic growth function:`f(x)=(11)/(121x-x^2)`
Please show work/explain :)

Answer 1

`11/((121-x)x) = 1/11 (1/x+1/(121-x))`

Explanation:

The given function is

`f(x) = 11/(121x-x^2) = 11/((121-x)x)`

We try a partial fraction decomposition of the form

`11/((121-x)x) = A/x+B/(121-x)`

Multiplying both sides by `(121-x)x` leads to

`11 = A(121-x) +Bx`

Since this is an identity, it is correct for all values of `x`. Substituting `x=0` gives

`11 = 121A implies A = 1/11`

Again, substituting `x=121` leads to

`11 = 121B implies B = 1/11`

Thus, the required decomposition is

`11/((121-x)x) = 1/11 (1/x+1/(121-x))`