Results 1 to 2 of 2

Thread: Showing that a function satisfies an ODE

  1. #1
    Jan 2011

    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 T_0, T_1 and H 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 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


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

    p=(kbT)/m, dp/dk=bT/m, 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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Mar 2010
    We have

    <br />
\displaystyle<br />
\frac{dp}{dx}=-gk \; \; (1)<br />


    <br />
\displaystyle<br />
p=ckT<br />

    we get

    <br />
\displaystyle<br />
k=\frac{p}{cT}<br />

    and inserting to (1) we have

    <br />
\displaystyle<br />
p=-g\frac{p}{cT}. \; \; (2)<br />

    Inserting temperature

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

    into (2) we get

    <br />
\displaystyle<br />
p=-g\frac{p}{c(T_0+T_1 \; e^{-x/H})}. <br />
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: Mar 14th 2011, 11:46 AM
  2. Show Function Satisfies Equation
    Posted in the Algebra Forum
    Replies: 6
    Last Post: Nov 25th 2010, 11:43 AM
  3. Replies: 1
    Last Post: Feb 26th 2010, 07:08 PM
  4. function that satisfies first 3 axioms
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: Sep 18th 2009, 09:00 AM
  5. Help me ! A function satisfies the differential equation
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: Feb 11th 2009, 11:28 AM

Search Tags

/mathhelpforum @mathhelpforum