Solving an ODE with parameters and conditions

Question

I am trying to use Mathematica to solve a relatively simple ODE involving parameter(s). I would like to use a set of conditions to solve for the particular solution of the ODE. I understand how to make Mathematica find values for the constants that arise during the process of solving the ODE, but what about solving for constants/coefficients already present in the original ODE? Here is a simple example involving Newton's Law of Cooling...

Here is the code I tried:

   DSolve[
     {
      T'[t] == -k*(T[t] - Ta),
      T[0] == 70,
      T[1/2] == 110,
      T[1] == 145
      },
     {T[t], t, k},
     {t}
     ]

I feel like I need a two step process... first solve the ODE with the parameters, and then solve for the parameters afterwards. I'm just not sure where to start.

Thank you in advance!

Answer 1

sol = T[t] /. First@DSolve[{T'[t] == -k*(T[t] - Ta)}, T[t], t]

sol = sol /. C[1] -> c

eq1 = 70 == sol /. t -> 0;
eq2 = 110 == sol /. t -> 1/2;
eq3 = 145 == sol /. t -> 1;
Solve[{eq1, eq2, eq3}, {k, c, Ta}]

Answer 2

DSolve[D[T[t], t] == -k*(T[t] - Ta), T[t], t]

{{T[t] -> Ta + E^(-k t) C[1]}}

At t = 0, (substituting t = 0 above), T[0] = Ta + C[1],

therefore C[1] = T[0] - Ta.

Substituting C[1] gives:

T[t] -> Ta + (T[0] - Ta) E^(-k t)

Given T[0] = 70

and rearranging the equation to the form: k == -(1/t) Log[(T[t] - Ta)/(T[0] - Ta)]:

Solve[{k == -2 Log[(110 - Ta)/(70 - Ta)],
  k == -Log[(145 - Ta)/(70 - Ta)]}, {k, Ta}]

{{k -> Log[64/49], Ta -> 390}}

Therefore k = 0.267063, Ta = 390 and c = T[0]-Ta = -320