Re: Methods to resolve this kind of differential equations
- From: David C. Ullrich <dullrich@xxxxxxxxxxx>
- Date: Wed, 14 Jan 2009 09:08:31 -0600
On Wed, 14 Jan 2009 07:06:31 EST, RM <ricardo.m.s.machado@xxxxxxxxx>
wrote:
Take the following differential equation:[i] y''(t).t^2 +4.t.y'(t)+2.y(t) = 1/t , t not null
What methods can one use to solve this sort of equation?
This sort of equation, who knows. Possibly power series
(or Laurent series).
For this particular equation, if this is not homework,
you could note that y_1 = 1/t is an "obvious" solution
to the related equation
[ii] y''(t).t^2 +4.t.y'(t)+2.y(t) = 0,
you could use "reduction of order" to find a second
independent solution to [ii] and then "variation of
parameters" to find the solution to [i]. (Actually
the two steps can be combined...)
Hmm. It's not clear how general "this sort of
equation" is supposed to be. If a, b, and c are
constants then you should be able to find one
or two solutions to
[iii] a t^2 y''(t) + b t y'(t) + c y(t) = 0
by assuming y = t^r and figuring out what r works;
if there's only one value of r that works then
t^r log(t) should be a second solution, and then
given those you can use variation of parameters on
a t^2 y''(t) + b t y'(t) + c y(t) = g(t).
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.
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