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For a species of bacterium, the rate of increase of the population is determined by the following information:

the rate of increase is proportional to the population P(t)

the rate of increase is proportional to the temperature T(t)

My question:

I initially wrote out two seperate equations for this, namely dP/dt = aP(t) and dP/dt = bT(t), then using the initial conditions P(0)=1000 and P(10)=2000 and the information T(t)=40-2t solved each, one giving an exponential and the other a polynomial.

Should I have instead solved the equation dP/dt = kP(t)T(t)?

I'm pretty sure the answer is 'yes' but I'd like some confirmation anyway.