Assuming “instantaneously compounded interest” the problem is of the form d P(t) = (r P - a ), where r is the rate of interest, and a is the rate of repayment.
The integral is number 27. in the CRC math tables, but we all know this anyway, as we realize as soon as we look it up :-)
From this formula I get
P1 = P0(1-a/(r P0) ) exp( r t ) - a/r
This seeming weirdness makes sense. In the first place, you must have your rate of repayment, a, greater than your instantaneous AMOUNT of interest, r P0, where P0 is your initial principal owed. Otherwise, you will make no progress whatsoever. So as to conceivability, the question is settled.
And since the progress is negative in the “meme” story, this must be the case there.
Suppose your interest is a usurial 10% annually. This would be $2640 annually which would have to be matched by a $220 monthly payment just to pay back the accummulating interest.
For the Principal to have “only” doubled in 23 years means that, if the story is substantially true, the complainant must have been paying a monthly amount which almost matched the monthly interest on the initial principal. Of course, this is a runaway situation once the principal grows substantially, and I imagine the initial intention was to “keep pace with the interest” with minimal payments.
s/b
P1 = P0(1-a/(r P0) ) exp( r t ) + a/r
That was a transcription error. I had as the condition for complete payment:
P0(1-a/(r P0) ) exp( r t ) = - a/r
which goofed me up. That little residual amount, a/r, which must be paid off according to the formula, is interesting mathematically. Note this is nearly P0 when a ~= rP0, and it shows that the defrayment of the principal is a small fraction of the interest payments when your payments barely cover the interest. I suppose we might have guessed that!