Linear differential equation

The name is a bit obscure if you don't think in very generalized terms right out of the gate. It refers to a linear polynomial of multiple variables, which by definition must have the super simple form of:

f (x_{0}, x_{1}, . . ., x_{n}) = c_{0} x_{0} + c_{1} x_{1} + . . . + c_{n} x_{n} + k

(20)

and then we just put the unknown

y

and each derivative into that simple polynomial:

f (y (x), y^{'} (x), . . ., y^{(n)} (x)) = c_{0} y + c_{1} y^{'} + . . . + c_{n} y^{(n)} + k

(21)

except that now the

c_{i}

are not just constants, but they can also depend on the argument

x

(but not on

y

or its derivatives).

Explicit solutions exist for the very specific cases of:

constant coefficients, any degree. These were known for a long time, and are were studied when Ciro was at university in the University of São Paulo.
degree 1 and any coefficient

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