Showing that a function satisfies an ODE

Hi, I have the following problem and I am stuck... Thanks for any help you can give me!

The problem says:

The atmospheric pressure p>0 decreases with height x according to the equation

where k is the air density, and g is the (constant) acceleration due to gravity. The density k is related to the pressure and temperature T by the ideal-gas law

where c>0 is a constant. This assignment examines the pressure distribution in an atmosphere under the assumption that the temperature decreases with height according to

where , and are positive constants.

Q1: Show that p satisfies the ODE

where L is a constant which you will relate to g and c.

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So here's what I understand and what I have done so far.

and , so I need to find a way to transform to ? What am I doing wrong?

I have seen other examples, but they do it in a different way. For example, they have

, (where b is a positive constant and m is the mass of a molecule of air). In one of the examples it says: if T is constant, show that

This makes more sense when it comes to transforming the to , since

, , and

My teacher usually mistypes things, but no one else has complained about it so I don't know if he's mistyped it or if I don't know how to do it.