Radomir: Here, the dative + infinitive shows necessity or obligation or lack of choice -- in other words, справиться одному has basically the same meaning as надо справиться одному: not simply "to do it alone", but "to have to do it alone" or "to be obliged to do it alone" or "to find it necessary to do it alone", etc.

So справиться одному с чем-нибудь трудным can be translated in English as "to be forced to struggle by oneself with something difficult."