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The only difference between the two phrases is in their formality level. どのように is more formal (and polite) than どうやって. If you are familiar with the verb やる, you know that it is a casual/conversational verb. やって is a form of やる. What is the verb that is more formal than やる but has the same meaning? It is する. Thus, you can also say どのようにして to mean ...
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