# Math Help - Showing that a function satisfies an ODE

1. ## 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

$dp/dx=-gk$

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

$p=c*k*T$

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

$T=T_0+T_1e^{-x/H}$

where $T_0$, $T_1$ and $H$ are positive constants.

Q1: Show that p satisfies the ODE

$dp/dx=-(L*p)/(T_0+T_1e^{-x/H})$

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.

$dp/dk=cT$

and $dp/dx=-gk$, so I need to find a way to transform $dp/dk$ to $dp/dx$? What am I doing wrong?

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

$dp/dx=-gk$, $p=kbT/m$ (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

$dk/dx=-((mg)/(bT))k$

This makes more sense when it comes to transforming the $dk/dx$ to $dp/dx$, since

$p=(kbT)/m$, $dp/dk=bT/m$, and

$dk/dx=(dp/dx)/(dp/dk)=-((mkg)/(bT))$

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.

2. We have

$
\displaystyle
\frac{dp}{dx}=-gk \; \; (1)
$

From

$
\displaystyle
p=ckT
$

we get

$
\displaystyle
k=\frac{p}{cT}
$

and inserting to (1) we have

$
\displaystyle
p=-g\frac{p}{cT}. \; \; (2)
$

Inserting temperature

$
\displaystyle
T=T_0+T_1 \; e^{-x/H}
$

into (2) we get

$
\displaystyle
p=-g\frac{p}{c(T_0+T_1 \; e^{-x/H})}.
$