dT/dt = e^(bt) − kT. The first term on the right side is time-varying. As a result, the differential equation is not separable. use change of variables

Question

mRNA is an intermediate molecule that arises in the decoding of DNA. mRNA is produced by a process called transcription and it eventually decays. Suppose that the rate of transcription is changing exponentially according to the expression e^bt and mRNA has a constant per capita decay rate of k. The number of mRNA transcript molecules, T, thus changes as

dT/dt = e^bt  − kT .

The first term on the right side is time-varying. As a result, the differential equation is not separable. However, the equation can be solved using the change of variables y(t ) = be^kt T(t ). Solve the differential equation using this technique. (Let T (0) = T₀.)

I do not really know how to approach this problem using separation of equations. Do you substitute the y(t) using change of variables? How do I know when to do this?

Answer 1

It is a linear equation:

dT/dt + kT = exp(bt)

The integrating factor is exp( integral(k)) = exp(kt)

Multiples everything by exp(kt)

dT/dt * exp(kt) + k*exp(kt)* T = exp(bt) * exp(kt) = exp( (b+k)t)

Notice that the left side is the derivative (by product rule) of:

d/dt ( T * exp(kt)) = exp ( (b+k)t)

integrating both sides with respect to time t....

T ( exp(kt)) = integral ( exp(b+k)t)

the right side integrates using the substitution U = (b+k)t ---> dU = (b+k) dt ---> dU/(b+k) = dt

T * exp(kt) = 1/(b+k) * exp((b+k)t) + C = 1/(b+k) * exp(bt+ kt) + C

multiplies everything by exp(-kt)

T = 1/(b+k) * [ exp(bt) + C * exp(-kt) ]