I am trying to find the trial solutions to the following differential equations. I would be able to solve them, but I am not sure how to go about simply determining trial solutions.

1)

2)

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- Sep 25th 2011, 10:30 AMchemeFinding trial solutions of 2nd order homogeneous DE.
I am trying to find the trial solutions to the following differential equations. I would be able to solve them, but I am not sure how to go about simply determining trial solutions.

1)

2) - Sep 25th 2011, 02:50 PMJesterRe: Finding trial solutions of 2nd order homogeneous DE.
What do you mean "trial solutions"?

- Sep 25th 2011, 06:37 PMchemeRe: Finding trial solutions of 2nd order homogeneous DE.
That is what I don't understand. I guess no one will be able to assist on this. I thought trial solution was something specific for certain types of problems.

- Sep 25th 2011, 07:52 PMJesterRe: Finding trial solutions of 2nd order homogeneous DE.
May we ask how you came upon the term "trial solution"?

- Sep 26th 2011, 01:38 AMchisigmaRe: Finding trial solutions of 2nd order homogeneous DE.
I'll try to find a solution of 1)... but I don't know if it is or not the 'trial solution'(Thinking)...

Using t as independent variable the DE is...

(1)

If is the Laplace Tranform of f(t), then a basic property is...

(2)

Now if we use the property (2) in (1) we obtain...

(3)

... that with some steps becomes...

(4)

... so that we have now a first order linear DE in s, of course more 'approachable' then (1). The (4) can be solved with 'standard procedure' obtaining...

(5)

At this point if You have a look at the manual, You discover that is...

(6)

... where is the Bessel function of first kind of order 0. Only one consideation: in (6) we have only one 'arbitrary constant' c even if the (1) is a linear ODE of order two. The [probable] reason is that the procedure finds only the solutions of (1) that are L-transformable, so that the other independent solution, probably with a singularity in t=0, is 'desaparecida'...

Kind regards

- Sep 26th 2011, 08:15 PMchisigmaRe: Finding trial solutions of 2nd order homogeneous DE.
Now we pass to...

(1)

... which is a little modified respect to the ODE 2) in the original post [1 instead of 3...]. Here we only want to show how to proceed in cases like this...

In order to eliminate the 'unconfortable' term , let's suppose that is . In thast case is...

(2)

(3)

... and, inserting (3) in (1), we have...

(4)

The (4) is identified as a Bessel differential equation of order and its solution is...

(5)

... so that the solution of (1) is...

(6)

Kind regards