Linear vs nonlinear diff equation II

1. Oct 20, 2015

Calpalned

For the linear differential equation, it says that a_n and b_n are constants or functions of x. This implies that a function of x = f(x) = constant. How can a number be a constant if it is a function? I think I'm misunderstanding something.

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2. Oct 20, 2015

Geofleur

They mean that $a_n$ and $b_n$ can be either functions of $x$ or constants, but not both. In other words, they are using "or" in the exclusive sense, not in the inclusive sense. We could also think of $f(x) =$ constant as a special case of a "function of $x$", albeit a trivial one. Think of a machine that spits out the same thing, no matter what you feed into it.

3. Oct 20, 2015

Calpalned

If I am not mistaken, a trivial solution is one where the solution is zero?

4. Oct 20, 2015

Geofleur

Well, trivial can also mean "not very interesting".

5. Oct 20, 2015

SteamKing

Staff Emeritus
Yes, the trivial solution is the zero solution.

6. Oct 21, 2015

HallsofIvy

The crucial point in "linear" equations is that the coefficients, $a_i$, cannot be functions of y or any of its derivatives.