Have a look to these examples

Dear Ghady, your first order ode seems to be linear non homogeneous, for which an analytic expression of solution exists. See e.g.

https://en.wikipedia.org/wiki/Linear_differential_equation

or the book on ODE by Boyce - Di Prima

Gianluca

Sharing link:

https://www.google.es/url?sa=t&source=web&rct=j&url=http://www.math.psu.edu/tseng/class/Math251/Notes-2nd%2520order%2520ODE%2520pt1.pdf&ved=2ahUKEwiW9-Gkz6zZAhWBNhQKHdzjA6QQFjABegQIDxAB&usg=AOvVaw2ngT2kkDD-zp4C0RCb7ob_

Regards!

The equation possesses irregular solutions, only, more precisely the solutions cannot be determined by their value at the instant t=0. Indeed, for the homogeneous part we have:

dT/T = - (1 + A(t)) dt /t , where A(t) = h S(t) / (m C_p)

which implies the following formula T(t) = C T_o(t), with

T_o(t) = \exp{ - \int A(t)/t dt ] } / t (for homogeneous part)

where C is an arbitrary constant. BUT, one cannot derive the value of C from the initial condition, unless A(t) is approximately equal to -1 as t goes to 0 (which sounds very restrictive). This feature needs an explanation about the kind of initial conditions used in the posted problem.

Next, the standard way suggested in the recommended links to "theoretical" algorithms, requires that T_o should be determined explixitly before the inhomogeneous solution is to be proceeded. Indeed, the usual formulas give us for the varying constant C(t) the following DE:

C'(t) = [ T_e + A(t) ] t T_o(t) = [ T_e + A(t) ] \exp{ - \int A(t)/t dt ] }

I am not familiar with the possibilities of MAPLE, but I think that in such case a better way would be to use any general numerical algorithm of solving ODE of the first order, without using the advantge of its linearity.

PS. Which type formulas are modelling the dependence of the surface S(t) on time?

Regards, Joachim