A didactic poem is one that teaches a lesson, either factual or moral, and often in a heavy-handed way. One thing that can make a poem "didactic" is to address the reader directly with imperatives, that is, it directly tells the reader to do something or to think something. There are other ways it can try to teach a lesson, but the teaching needs to be overt and aboveboard for the poem to be called "didactic." 
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The psalms have long been recognized as poetry,i.e., text whose intellectual content is expressed in imaginative and emotional language.
They impart more than knowledge.
They impart imaginary and sentiments.