Question

DIFFERENTIAL EQUATIONS
Find the function f s.t. f'(x) = f(x)*(1- f(x)) and f(0) = 1/2. Does this want me to solve for y, if I say y = f(x)?

I show you the eqn I used:

Answer 1

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Answer 2

\[\frac{dy}{dx} = y(1-y)\]
Where y = f(x)

Answer 3

@saifoo.khan are you experienced with this?

Answer 4

This wants me to solve for y =, right?

Answer 5

But here is the problem:
\[\ln|1 - y| + \ln|y| = x + \ln(1/4)\]
How the heck can you solve for y there?

Answer 6

Introduce ln to every term then differentiate.
Derivative of \[\ln x= 1/x\]
But then we have y. So remember to multiply with y' or dy/dx then solve for y' .Need me solve it?

Answer 7

The absolute doesn't make a difference because ln x has to be greater than zero

Answer 8

If it says find the function f (or I said f(x) = y), does that not mean I need y =?

Answer 9

Sorry no.

Answer 10

Hold on, So what is the function you wrote down equal to?

Answer 11

you have to find f(x),
\[f'(x) = f(x)(1- f(x)) ~and ~f(0) = 1/2\]
\[\frac{dy}{dx}=y(1-y)\]
\[\begin{array}{l} \text{rearrange} \\ \frac{dy}{dx}\text{ = }-(y-1) y \\ \text{divide both sides by }-(y-1) y \\ -\frac{1}{(y-1) y}\frac{dy}{dx}\text{ = }1 \\ \text{integrate both sides } \\ \int\limits -\frac{1}{(y-1) y} \frac{dy}{dx}\, dx\text{ = }\int\limits 1 \, dx \\\end{array}\]
you'll get
\[-\ln (-y+1)+\ln (y)\text{ = }x+c\]

Answer 12

WTH? Where did you get that (-) from?

Answer 13

you rearrange it

Answer 14

Ok I see it

Answer 15

Oh so you use the quotient rule for logs afterwards to get rid of the possibility of there being a y^2 term

Answer 16

U substitution @QRAwarrior That's what you use to integrate such a function.

Answer 17

\[-\ln (-y+1)+\ln (y)\text{ = }x+c\]
\[\ln(\frac{y}{1-y})=x+c\]
\[\frac{y}{1-y}=e^{x+c}\]

Answer 18

Wait, so then you do NOT get a "y =" expression, because of that (y)/(1 -y)..

Answer 19

\[\frac{y}{1-y}=e^{x+c}\]
\[y=e^{x+c}(1-y)\]
\[y=e^{x+c}-ye^{x+c}\]
\[y+ye^{x+c}=e^{x+c}\]
\[y(1+e^{x+c}=e^{x+c}\]
\[y=\frac{e^{x+c}}{1+e^{x+c}}\]

Answer 20

Ok I do not want to look at that to its entirety until I try your rearranging strategem. Thanks for your help.

Answer 21

yeah you should try rearranging them, then sub the f(0) = 1/2 and get c, then present the function with y and x