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.
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.