Find the function f such that  `f'(x) = f(x)(1-f(x))` and `f(0) = 1/2`  

Expert Answers
embizze eNotes educator| Certified Educator

Given `f'(x)=f(x)(1-f(x))`

and `f(0)=1/2:`

Let y=f(x) and rewrite as:


Use separation of variables to get:


Rewrite the left hand side using partial fractions:

`(dy)/y - (dy)/(1-y) = dx`


`int (dy)/y - int(dy)/(1-y)=int dx`






`1/y=ce^(-x)+1 `

`y=1/(ce^(-x)+1) `

`y=(e^x)/(c+e^x) `

At 0, `y=1/2 so`

`1/2=1/(c+1) ==> c=1`



embizze eNotes educator| Certified Educator

The answer is correct. The partial fraction expansion is not written correctly.


so we would have `dy/y + dy/(1-y)`

but the derivative of 1-y is -dy so the integrals should be written as:

`int dy/y - int (-dy)/(1-y)=int dx`

and the rest will follow.

You can check the answer:




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