The word desolate hasn’t strayed far from its Latin roots: its earliest meaning of “deserted” mirrors that of its Latin source dēsōlātus, which comes from the verb dēsōlāre, meaning “to leave all alone; forsake; empty of inhabitants.” That word’s root is sōlus, meaning “lone; acting without a partner; lonely; deserted,” source too of the “lonely” words sole, soliloquy, solitary, solitude, and solo. Desolate also functions as a verb (its last syllable rhymes with wait rather than what) with its most common meanings being “to lay waste” and “to make wretched; to make someone deeply dejected or distressed.”
Middle English desolaten (in past participle desolatid "deserted, ruined"), borrowed from Latin dēsōlātus, past participle of dēsōlāre "to leave all alone, forsake, empty of inhabitants" — more at desolate entry 1
Middle English desolat "having no inhabitants or visitors, deserted," from Latin desolatus, past participle of desolare "to abandon," from de- "from, away" and solus "alone" — related to soleentry 4, solitude, solo