I would like to know how do we analyze or regard the Neutral response (such as Don't Know) in a questionnaire using the following Likert Scale:
1:Strongly Disagree 2:Disagree 3:Don't Know 4:Agree 5: Strongly Agreee
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"Don't know" is a bad neutral option, because it's not really neutral – it's barely an answer at all. If you are short on data, I suggest treating it as missing. You can then test for systematic missingness here much as you could with any other missing data. Responses that use the other four rating options are more properly ordinal, so you can equate them with ranks 1–4.
If you have a lot of data and at least a few items with this rating scale, you can try item response theory analyses. For instance, if you want to estimate a latent continuous variable using many items that have this same rating scale, a rating scale model would help you determine whether "don't know" contains any information about the latent variable at all. My guess is that it wouldn't, but if you have the data and the patience to study the methodology, you can make an empirical question and answer of it.
In general, I would recommend against including "Don't know" as the neutral option in your Likert scale. In doing that, you're not differentiating between two types of people: 1) those who really don't know or have an opinion (they may be unfamiliar with the topic, or refuse to answer), and 2) those who feel neither positively nor negatively about the subject at hand. These types of people could be very different. In fact, in a lot of surveys, people will include a neutral scale midpoint such as "Neither agree nor disagree" and also another separate option to select "Don't know" which would be treated as a nonresponse. If you want to read more about these issues I recommend looking at this resource:
Krosnick, J.A. and L.R. Fabrigar. 1997. Designing rating scales for effective measurement in surveys. In: (L. Lyberg, P. Biemer, M. Collins, E. de Leeuw, C. Dippo, N. Schwarz and D. Trewin, eds.) Survey measurement and process quality. John Wiley and Sons, Inc., New York, NY. pp. 141–164.