My dependent variable should be binary, but can I include discrete and continuous variables simultaneously in my equation? For example let's say X1 is discrete and X2 is continuous?
There is absolutely no problem, just code your categorical predictor(s) as dummy variables, or some other form of categorical-encoding. This can be used with all form of regression models. It is usually the type of the response ($Y$) variable which "dictates" the type of regression model that can be used, not the predictors.
So, for a binary response, logistic regression, for a multinomial response, multinomial logistic regression, continuous response, muliple linear regression, and so on (there are of course alternatives). But in these decisions the type of predictor variable generally plays little role. See for instance Model for continuous response and a mix of continuous and categorical predictors and Predicting with both continuous and categorical features