It depends on which gene or sequence you are amplifying. Compare the two sequences from P. putida and P. aeruginosa to see how similar or different they are and then determine where the specificity of the primers for each. It could be that you need to design different primers, or it could be that your PCR will be able to discriminate between the two strains by changing the PCR conditions, such as higher annealing temperature.

i amplified P. aeruginosa DNA with algD GDP mannose dehydrogenase gene primer which is mentioned in literature as specific for P. aeruginosa but its amplifying P. putida DNA also.

I need to species specific primer to pseudomonas stutzeri?