Linear vs nonlinear diff equation II

[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.

 

[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.

 

[Calpalned]

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.

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

 

[Geofleur]

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

 

[SteamKing]

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

Yes, the trivial solution is the zero solution.

 

[HallsofIvy]

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