Is it obligatory for me to divorce my wife to appease my mother, who disapproves of my marriage before my older sister, or should I continue my marital life with the public declaration of the contract despite my mother's anger?

You are not obligated to divorce your wife in response to your mother's command, especially if her request is without a valid reason, but merely due to whim. Divorcing your wife is not an act of showing dutifulness to your mother. You must strive to be dutiful and benevolent towards your mother.