LotR -- Book 5 - Chapter 02 - The Passing of the Grey Company

[Celuien]

'Too often have I heard of duty,' she cried. 'But am I not of the Houseof Eorl, a shieldmaiden and not a dry-nurse? I have waited on faltering feet long enough. Since they falter no longer, it seems, may I not now spend my life as I will?'
'Few may do that with honour,' he answered.

Theoden seems to have strongly impressed duty onto Eowyn. With the high value placed on duty and honor, he would have had no reason to think that Eowyn would abandon her charge, even if te thought had crossed her mind. I agree that her belief that Aragorn was going to his death helped spur her on to the decision; perhaps it was also related to the realization that Aragron did not return her love.

At any rate, it was fortunate that she did finally choose to leave Edoras.

[Kuruharan]

Theoden seems to have strongly impressed duty onto Eowyn.

This is true. But to some extent I think he may have been putting feelings and impulses onto Eowyn in his own mind that were inaccurate (this is a common problem).

He does not seem to entirely understand the impact all those years of taking care of him and listening to Wormtongue has had on her.

[Formendacil]

If Eowyn is selfish in abandoning her responsibility, and wishing to go off to war and glory, is Theoden not equally guilty? After all, as both an old man and the king of Rohan, it would be more natural for him to remain at home, instead of Erkenbrand, and send the younger marshals out to command the army. Instead, he wants one last gasp at glory.

Now, I admit that Theoden had some more altruistic reasons than that- such as the morale it gives his men, but to Eowyn it might appear as such.

Perhaps Eowyn is following her uncle's example: first she does her duty, as does he, then she goes chasing after glory, as does he.