Differential equation problem and modulus sign

[zsteve]

Hi, I have the following question:
The rate at which a rumour is spread through a year level satisfies where R(t) is the number of students which have heard the rumour after t minutes. Given that R(0)=5, show that
.
My question is: in solution of this differential equation, I will end up with the following:
.
From the question text, I cannot deduce whether the expression inside ln(...) on the LHS is positive or negative, as I cannot show that 50-R is positive (although if we assume it is, everything works out fine).

So my question is, is there some convention where we just assume it is positive and get on with it, or do we have to say it is equal to when getting rid of the log?

And you can't actually infer it is positive from the text, can you?

Just checking

[cosine]

Hi zsteve,

Well you could do the if you wish, which is what the book probably does, but then they deduce the negative because you simply cannot have, in this case, negative students that have heard the rumour. So essentially instead of always putting the and then deducing the negative answer, you can just list the positive one.

Hope that makes sense

[keltingmeith]

R(0)=5 means that at some point, there are 5 students who know the rumour. This means that 50-R *must* be positive.

This initial condition determines if you can make the log positive or negative.

[zsteve]

@Cosine & EulerFan101 : aha thanks.
@EulerFan101: what about differential equations that aren't based on a physical situation? For example, the general solution to a linear first order ODE. When solving for the integrating factor, it's always said to be positive. However, if you use modulus, it's plus/minus. Why do they do this?

[cosine]

@Cosine & EulerFan101 : aha thanks.
@EulerFan101: what about differential equations that aren't based on a physical situation? For example, the general solution to a linear first order ODE. When solving for the integrating factor, it's always said to be positive. However, if you use modulus, it's plus/minus. Why do they do this?

Im not EulerFan, however from personal experience, the textbooks might not include the plus or minus and just include the positive solution. But even if you look at checkpoints, to gain the full marks you would have needed to include both the positive and negative values. So in this case, the error lies in our textbooks.

It's also similar like when you solve for x:





Okay well, it's sort of different, but in essence you're still solving for something, and if it's just a normal general solution, you must include both signs as they are both solutions.

[keltingmeith]

@Cosine & EulerFan101 : aha thanks.
@EulerFan101: what about differential equations that aren't based on a physical situation? For example, the general solution to a linear first order ODE. When solving for the integrating factor, it's always said to be positive. However, if you use modulus, it's plus/minus. Why do they do this?

Integrating factor is different because that gets fixed when you eventually solve the DE.

If it's not based on a physical situation, it doesn't matter - the log has to have a positive argument. We're not looking for 50-R to be positive because we need a positive amount of people, we're looking for 50-R to be positive so that the log is defined.  As cosine is sort of alluding to, the only time you include the +/- is if you don't have initial conditions, and so cannot make further judgements.