$$\frac{dT}{dt} = - k(T-T_a)$$ Where T(t) is the temperature of the object at time t, $T_a$ is the ambient temperature and k is a positive constant.
I know this can be easily solved by the separable variable method however, I’m trying to solve it using the integrating factor method which has to be in the form of $\frac{dy}{dx} + p(x) \cdot y = f(x) $
is this possible? I heard that it is possible with assumptions but I am not sure how to go with it.
Expand and move terms to get into standard form $$\frac{dT}{dt} + kT = kT_0$$
Your integrating factor is $e^{whatever\, is\, next\, to\, T}$ in this case the k is next to T
$$e^{\int k dt} = e^{kt}$$
Multiply everything by integrating factor
$$\frac{dT}{dt}e^{kt} + kT e^{kt} = kT_0e^{kt}$$
Separate out the derivative
$$\frac{d}{dT}[Te^{kt}] = kT_0 e^{kt}$$
Integrate both sides $$Te^{kt} = \int kT_0 e^{kt} dt$$
$$Te^{kt} = kT_0 \frac{e^{kt}}{k} +C$$
Divide both sides by $e^{kt}$ and simplify
$$T = \frac{kT_0 e^{kt}}{k e^{kt}} + Ce^{-kt}$$ $$T = T_0 + Ce^{-kt}$$