Separating Variables in Differential Equations: Solving dP/dt = P - P^2

[bdh2991]

Homework Statement

dP/dt=P-P^2

Homework Equations

The Attempt at a Solution

I know you can separate this and after i did that and did my partial fractions i got

t + C = ln(P) + ln(1-P) but i don't know what to do from here i figure you take the e of both sides at some point but i never ended up with the right andswer please help

 

[LCKurtz]

Homework Statement

dP/dt=P-P^2

Homework Equations

The Attempt at a Solution

I know you can separate this and after i did that and did my partial fractions i got

t + C = ln(P) + ln(1-P) but i don't know what to do from here i figure you take the e of both sides at some point but i never ended up with the right andswer please help

I think you have a sign error in your solution. You need to fix that and remember the formulas like ln(a)+ln(b) = ln(ab) and ln(a) - ln(b) = ln (a/b).

 

[bdh2991]

i'm not seeing where the sign error is coming from...for my partial fractions i did

A/P + B/(1-P), then getting rid of the denominators i get A(1-P) + Bp = 1

solving for A and B i get A=1, B=1

so wouldn't it be ln(P) + ln(1-P)??

the only thing i could think is that it should be A(1-P) - BP

but i don't really understand how it could come out as a subtraction

 

[BrownianMan]

i'm not seeing where the sign error is coming from...for my partial fractions i did

A/P + B/(1-P), then getting rid of the denominators i get A(1-P) + Bp = 1

solving for A and B i get A=1, B=1

so wouldn't it be ln(P) + ln(1-P)??

the only thing i could think is that it should be A(1-P) - BP

but i don't really understand how it could come out as a subtraction

Your partial fraction should be 1/P - 1/(P-1).

 

[LCKurtz]

i'm not seeing where the sign error is coming from...for my partial fractions i did

A/P + B/(1-P), then getting rid of the denominators i get A(1-P) + Bp = 1

solving for A and B i get A=1, B=1

so wouldn't it be ln(P) + ln(1-P)??

Your sign is wrong on your integration of$$
\int\frac 1 {1-P}\, dP$$

 

[bdh2991]

I see i was messing up my U-substitution, it's always the little things lol thanks for the help