Find the function f such that `f'(x) = f(x)(1-f(x))` and `f(0) = 1/2`
Expert Answers
| Certified Educator
briefcaseTeacher (K-12)
calendarEducator since 2011
write3,179 answers
starTop subjects are Math, Science, and Business
The answer is correct. The partial fraction expansion is not written correctly.
`1/(y(1-y))=1/y+1/(1-y)`
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:
`y'=e^x/(1+e^x)^2`
`f(x)(1-f(x))=(e^x/(1+e^x))(1-e^x/(1+e^x))`
`=e^x/(1+e^x)^2`
Related Questions
- If f(x)=x+4 and h(x)=4x-1, find a function g such that g(f(x)) = h(x).
- 2 Educator Answers
- `f'''(x) = cos(x), f(0) = 1, f'(0) = 2, f''(0) = 3` Find `f`.
- 1 Educator Answer
- Find a function f that satisfies the conditions: f " (x) = X^2 + X -1, f ' (0)=5, f (0)=10
- 1 Educator Answer
- Find the inverse function of f(x)=x^3+2?
- 1 Educator Answer
- f(x) = (x-3)/(x+1) find f'(0)f(x) = (x-3)/(x+1) find f'(0)
- 2 Educator Answers
| Certified Educator
briefcaseTeacher (K-12)
calendarEducator since 2011
write3,179 answers
starTop subjects are Math, Science, and Business
Given `f'(x)=f(x)(1-f(x))`
and `f(0)=1/2:`
Let y=f(x) and rewrite as:
`(dy)/(dx)=y(1-y)`
Use separation of variables to get:
`(dy)/(y(1-y))=dx`
Rewrite the left hand side using partial fractions:
`(dy)/y - (dy)/(1-y) = dx`
So
`int (dy)/y - int(dy)/(1-y)=int dx`
lny-ln(1-y)=x+c
`ln(y)/(1-y)=x+c`
`y/(1-y)=ce^x`
`(1-y)/y=ce^(-x)`
`1/y-1=ce^(-x)`
`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`
so
`y=e^x/(1+e^x)`