Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

First order homogenous DE with variable coefficient

  1. May 27, 2009 #1
    I was stumped by this differential equation. The function x = x(t).

    [tex]x^{\tiny\prime\prime} + \frac{1}{t}x^{\tiny\prime} = 0[/tex].

    You have the initial values x(a) = 0 and x(1) = 1.

    What I did was to introduce a new function u = x', so I ended up with the first order homogenous DE:
    [tex]u^{\tiny\prime} + \frac{1}{t}u = 0[/tex].

    This can't be that difficult, but the book which I am using does not give any details how to solve this particular problem. I only know how to solve it if I have a constant, or a function of t, on the right side.

    The solution to the problem is:
    [tex]x(t) = 1 - \frac{\log t}{\log a}[/tex].

    Hope someone can shed some light on this. Thank ye, ye scurvy landlubber! Yarrr!
    Last edited: May 27, 2009
  2. jcsd
  3. May 27, 2009 #2
    When you did a reduction of order, you arrived to a separable equation that isn't difficult to solve at all. There is no need to reduce order since the original equation can be rewritten in a simpler form. What happens if you multiply the second order ode by t?. Can you simplify it (i.e. the derivative of something)?
  4. May 27, 2009 #3
    Yes! Of course!! How could I miss that? :)

    Thank you for helping me!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook