# WitCH 138: Second Rate

Not counting the Witchfests on Nelson‘s Proof and Complex chapters, I don’t think we’ve had a textbook WitCH for quite a while. This one is hardly in the same league but it’s been on my to do list for a long time and it’s definitely a source of irritation. The topic came up in discussions with a colleague today, and I’m too busy and too tired to do the posts I should be doing, so this’ll do for now. It is an example from the differential equations chapter of Cambridge’s Specialist Mathematics 3 & 4, and it is related to this previous WitCH from the same text and chapter. As was the case with the previous WitCH, it is a representative, one of a type.

## 10 Replies to “WitCH 138: Second Rate”

1. Anothermouse says:

x = 0 when t = 0 is used as the boundary condition but x = 0 is excluded from the domain. Perhaps a limit should be used.

1. marty says:

Why is x = 0 excluded? In any case the endpoint thing is a fussy aspect that VCE never does well. Not one of my major concerns here.

1. Anothermouse says:

From the DE dx/dt = …. It’s not defined when x = 0.

1. marty says:

Excellent point. I missed that.

2. Simon says:

I guess they really want a DE for not

Of course, they make an overly big show of deriving and solving it too…
Much easier to use similarity on the volume instead of the side:

Then get the DE from the chain rule

Integrate up as usual to get their answer.
etc

1. marty says:

Thanks, Simon. Your similarity thing is nice. But even going directly, I think the text’s derivation is a mess. And good point about “differential equation for dx/dt”. I hadn’t noticed, but that is sloppy.

1. Simon says:

Funny! I thought your second rate title referred to the “DE for dx/dt” statement…

And yeah, the rest of their derivation is “Flippinâ€™ Ridiculous” – they should just treat the DE as a separable one / use a substitution to integrate from 0 to t in one clean move.

use the initial condition to get and then solve for .

1. marty says:

Yes, this section comes after Separation. So even on the text’s own silly terms, the flipping here is ridiculous.

3. John Kermond says:

Perhaps the solution should come with the disclaimer that it’s only valid for

.

Otherwise it leads us to believe that x increases without bound. As Dirty Harry said when he saw this question: "A model's got to know its limitations."

1. marty says:

The solution is valid for t = 0 as well, even if the meaning and the derivation needs to be clarified.