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Non-linear partial differential equations. Are you offering to help me? If not, why did you ask?
Equations are not linear when they are quadratic equations which are graphed in the form of a parabola
I'm not altogether clear about what you mean. However, the term 'linear programming' means a category of optimisation problems in which both the objective function and the constraints are linear. Please see the link.
They are not. A vertical line is not a function so all linear equations are not functions. And all functions are not linear equations.
Linear equations are a small minority of functions.