Why Does My RL Circuit Analysis Yield a Positive Time Constant?

[yosimba2000]

Homework Statement

After a long time, an inductor is disconnected quickly from the battery and then attatched to a resistor.

Homework Equations

V = Ldi/dt
V = ir

The Attempt at a Solution

KVL: voltage across inductor - voltage across resistor = 0
Ldi/dt - ir = 0
di/i = rdt/L
ln|i| from i(t) to i(0) = rt/L + C
i(t)/i(0) = e^{(rt/L +C)}
i(t) = i(0)Ce^rt/L

At time t= 0, i(t) = initial current i(0). Plugging in i(t) = i(0) and t = 0 gives C =1.

Now we have i(t) = i(0)e^rt/L

The correct equation is i(t) = i(0)e^-rt/L, different from what I have; I have a positive time constant r/L.

I know i(t) must be smaller than i(t) when time increases, and usually this is taken care of by a negative constant of integration, but as C = 1, the only way I can account for i(t) < i(0) is to use negative time.

Derivations I have seen started off KVL with -Ldi/di - ir = 0, but I don't understand why my method doesn't also work.

Is my method incorrect?

 

[kuruman]

After a long time, an inductor is disconnected quickly from the battery and then attatched to a resistor.

Please state the problem fully including what you are asked to find.