From the attachment it is clear that you have a couple of first order ODE with initial conditions 𝑇𝑎𝑖𝑛= 𝑇𝑎(0) and 𝑇𝑝𝑖𝑛 = 𝑇𝑝(0) . Solve them Runge Kutta method and find the solutions to obtain Ta0 and Tpout and use them in ∆𝑻 

Thank you very much.