Solving an algebraic equation...

[Random Variable]

How would you solve the following equation for P?

ln[(P-5)/P] +5/P = 25t +C

EDIT: The original problem is the following:

Let P(T) be the fish population in hundreds after t years. Assume that fish births occur at B=P^2 and that fish deaths occur at D=5P.

1) Set up the differential equation for this model and solve it. (The equation that has to be solved is dP/dt = (P^2 - 5P)*P, right?)

2) If there are initially 800 fish, will the fish population ever approach infinity?

I can't answer the second part if I can't solve ln[(P-5)/P] +5/P = 25t +C for P, right?

 

[silverpig]

1. Raise e to the power of that equation...

 

[Random Variable]

Originally posted by: silverpig
1. Raise e to the power of that equation...

y*e^(y) = e^(x+2)


Now what?

 

[MiataGirl]

y*e^(y) = e^(x+2)

divide by e^y now..

 

[KMurphy]

y=lambertw(e^(x+2))

 

[MiataGirl]

hmm, now that i try it it doesn't really seem possible..at least with any methods that i know of

 

[Random Variable]

Originally posted by: MiataGirl
hmm, now that i try it it doesn't really seem possible..at least with any methods that i know of

I would leave it as is, but I need to solve the equation for y(x) to do the second part of the problem.

EDIT: Actually, what I really want to solve is:

ln[(P-5)/P] +5/P = 25t +C

But the equation I gave is similar.

 

[RaynorWolfcastle]

Originally posted by: KMurphy
y=lambertw(e^(x+2))

Maple7 agrees

 

[RaynorWolfcastle]

Originally posted by: Vespasian

Originally posted by: MiataGirl
hmm, now that i try it it doesn't really seem possible..at least with any methods that i know of

I would leave it as is, but I need to solve the equation for y(x) to do the second part of the problem.

post the problem, you've got me intrigued

 

[KMurphy]

Originally posted by: icecool83

Originally posted by: KMurphy
y=lambertw(e^(x+2))

Maple7 agrees

As does Matlab 6.5

 

[Random Variable]

Originally posted by: icecool83

Originally posted by: Vespasian

Originally posted by: MiataGirl
hmm, now that i try it it doesn't really seem possible..at least with any methods that i know of

I would leave it as is, but I need to solve the equation for y(x) to do the second part of the problem.

post the problem, you've got me intrigued

Let P(t) be the fish population of a pond in hundreds after t years. Assume that the birth rate of the fishes = P^2 and that the death rate = 5P.

1) Set up the differential equation for this model and solve it.

2) If there are initially 800 fish in the pond, will the fish population ever approach infinity?

 

[Random Variable]

Originally posted by: KMurphy
y=lambertw(e^(x+2))

I don't know how to work with such an equation.

 

[JSSheridan]

I used to know how to do that. It's been so long since I had differential equations. It seems pathetic that a senior in Elec. Engineering can't do that. On the other hand, I've never had that problem like that outside of the math department. Anyway, good luck man.

 

[RaynorWolfcastle]

Originally posted by: Vespasian

Originally posted by: icecool83

Originally posted by: Vespasian

Originally posted by: MiataGirl
hmm, now that i try it it doesn't really seem possible..at least with any methods that i know of

I would leave it as is, but I need to solve the equation for y(x) to do the second part of the problem.

post the problem, you've got me intrigued

Let P(t) be the fish population of a pond in hundreds after t years. Assume that fish births occur at beta = P^2 and that fish deaths occur at delta = 5P.

1) Set up the differential equation for this model and solve it.

2) If there are initially 800 fish in the pond, will the fish population ever approach infinity?

Hmmm... try using linear algebra, it'll probably depend on initial population. (btw are beta and delta supposed to get plugged into an eqn? death rate = 5P seems odd)

 

[NogginBoink]

Originally posted by: JSSheridan
I used to know how to do that. It's been so long since I had differential equations. It seems pathetic that a senior in Elec. Engineering can't do that. On the other hand, I've never had that problem like that outside of the math department. Anyway, good luck man.

LOL... my brother is in the Army, in artillery. When deciding where to point the big guns, they not only use a fire control computer, but back it up with manual calculations.

He said to me, "I had to go up to my commanding officer, and in a somewhat shamed voice, admit, 'you know all those geometry and algebra courses I took where I told the teacher I'd never use this stuff in real life? I guess I'm going to have to take all that back now.'"

 

[NogginBoink]

doh... clicked wrong button.

 

[Random Variable]

btw are beta and delta supposed to get plugged into an eqn? death rate = 5P seems odd)

dP/dt = (birth rate - death rate)*P

 

[Random Variable]

bump

 

[silverpig]

Differential equations WAS easy for me back when I took that class. Now I can't remember sh!t.

 

[RaynorWolfcastle]

Since I'm a lazy bastard and didn't feel like doing it by hand, I plugged it into Maple's dsolve command and got
P(t) = exp(RootOf(_Z*exp(_Z)+5*_Z-25*t*exp(_Z)-125*t+5-ln(exp(_Z)+5)*exp(_Z)-5*ln(exp(_Z)+5)-25*_C1*exp(_Z)-125*_C1))+5

so ummmm... have fun with all of that

 

[Random Variable]

Originally posted by: icecool83
Since I'm a lazy bastard and didn't feel like doing it by hand, I plugged it into Maple's dsolve command and got
P(t) = exp(RootOf(_Z*exp(_Z)+5*_Z-25*t*exp(_Z)-125*t+5-ln(exp(_Z)+5)*exp(_Z)-5*ln(exp(_Z)+5)-25*_C1*exp(_Z)-125*_C1))+5

so ummmm... have fun with all of that

I think I'll pass.