Matlab; Partial Differential Equation; ODE - matlab

I am currently trying to solve a Matlab problem. For the first part, I need to hand solve Partial Differential Equation with initial and boundary condition. However, I am lost because I don't know how to continue solving the problem after I find the equation for finding T. There is an ODE inside the boundary condition, and with that ODE, there are another initial condition. How can I link everything together? I have also attached with my work as well. Please give me some idea, thank you.enter image description hereenter image description here

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How to set up a large set of first order non-linear differential equations in Matlab?

I am relatively new to writing my own code for Matlab though I have used the program a decent amount. Right now I am attempting to code a series of first order non-linear differential equations. They are all in one of two forms like the equations here:
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