Linear vs Nonlinear ODEs: What's the Difference and How Can You Analyze Them?

[Noesis]

I'm just curious as to what the actual distinction means.

I understand that the requirement for a linear ODE, is for all the coefficients to be functions of x (independent variable), and that all derivatives or y's (dependent variable) must be of degree one, but that doesn't tell me much.

In a normal function, there is a clear distinction between a linear and a nonlinear one.

For example, y = 3x, it's clear here that y changes linearly with x, and is always three times as big as x.

On y = x^2, it's obvious that the change is not linear...so the relationship isn't linear.

Now how can I analyze linear differential equations and nonlinear differential equations in a similar manner?

How does their behavior, or their 'meaning' differ?

 

[Euclid]

A linear function is one such that f(x+y)=f(x)+f(y). Think of a differential equation as a function. E.g.

f''+3x f' + 4 = 0

This can be thought of as a function F such that F(f)=f''+3x f' + 4. Then F is linear since F(f+h)=F(f)+F(h). The equation you want to solve is F(f)=0.

 

[arildno]

"f'' + 3x f' + 4 = 0"

This is NOT a linear diff.eq, since a sum of solutions isn't a solution.

 

[HallsofIvy]

arildno, by any definition I've seen, that would be a non-homogeneous linear differential equation. You are correct that the left-hand side is not "linear differential operator" but that's because it is not a differential operator at all- the "4" does not act on f. The equation can be rewritten f"+ 3xf'= -4 and now the left-hand side is a linear differential operator and the equation is a non-homogeneous linear differential equation.

I suspect Euclid meant f"+ 3xf'+ 4f= 0 and made a typo.