I have a very small data set, with two groups, 5 cases in each one. I ran the t-test, and got a p value of 0.049. The F test that SAS ran in addition gave a p value of 0.72. This is not a surprise, even if the variances differ, they need to differ "a lot" for the test to be significant with 5 cases per group. So, I tried ignoring the F test and looked at the t-test with Satterthwaite approximation of the degrees of freedom, and guess what ? The p value is 0.051. What should be my conclusion ? How to I decide in such a case ? The mean of group 1 is 34 and of group 2 is 12. The standard deviations are 16.4 and 13.5 respectively. By the way, I tried running it again in Stata, and it allows me to ask for the Welch approximation of degrees of freedom, the p value in this case was 0.043, but I read in an old post here that the Satterthwaite is more common in use.
You have equal sample sizes - the t-test is quite insensitive to the equal-variance assumption in that case.
(The standard deviations are also very close; the equal variance t-test is not that sensitive to mildly unequal variances even when the samples sizes aren't so close to equal.)
There should be no problem with using either test. The two tests should tell you about the same thing ... and in fact they do.
So there's no problem with using the equal variance t-test.
The only problem with what you did is not making the choice of which test you were using before seeing the results. That is problematic - if you are choosing between tests post-hoc your test procedures certainly won't have their nominal properties.