In German, the fruits you would put in a fruit salat are called Obst, in contrast to Frucht (fruit) / Früchte (fruits) which can be ‘anything’ complying with the botanical definition. You’d refer to tomatoes and paprika as Frucht-Gemüse (fruit vegetables).
It is not completely excluded. For cases where the ‘standard’ treatment did not work or does not exist, placebo treatment can be used under the condition, that the patient is informed that they will be given placebo.
Yes, and what is aimed for is separation of variables to find a solution of this differential equation. (The differential operator d should be printed as upright letter btw)