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OrdinaryDifferentialEquation

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Description

Summary

The first-order linear ordinary differential equation, solved by its integrating factor.
#241

Remarks

For y' + f(x) y = g(x), multiplying through by m = e^(int f dx) makes the left
side the derivative of m y — which is the whole method, since integrating both sides
then gives m y = int (m g) dx. Nothing here is a heuristic: where the two integrals
come out, the answer is exact, and where either does not this declines.
The unknown is an Application. A bare Variable cannot
stand for a function of x: derivative(y, x) is 0, because y does
not contain x and the library is right about that. Written apply(y, x) it is
an application that does not reduce, and its derivative stays symbolic — which is exactly
what an equation about an unknown function needs.
Linearity is checked, not assumed. The coefficients are read off by differentiating
with respect to the unknown and its derivative, which is only valid if the equation really
is linear in them — so the reading is put back together and compared with what it came
from. An equation that is not linear fails that comparison and is declined rather than
answered wrongly.

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