First order homogenous DE with variable coefficient

  • Thread starter MrGandalf
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  • #1
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Main Question or Discussion Point

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!
 
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Answers and Replies

  • #2
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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)?
 
  • #3
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Yes! Of course!! How could I miss that? :)

Thank you for helping me!
 

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