Understanding Non-Linear Differential Equations: Definition and Examples

[StationZero]

Hey gang, how would you specifically define what a non-linear DE is? I mean, what specifically qualifies a DE to be nonlinear, and what does that mean? I am currently working with a team that models brain function as an array of coupled oscillators with complex feedback profiles and I understand the the DE's that are put in the Runge-Kutta solver are nonlinear, but looking at a symbolic equation on a chalkboard, I'm not sure if I could pick out a non-linear ODE from a linear one. Could someone help?

 

[Unstable]

Think on what you learned at school. A linear function (really a straight line) is of the form
$y= a\cdot x + b$
Now a differential equation of the form
$x'(t) = A(t) \cdot x(t) + b(t) \qquad \qquad (*)$
is a linear differential equation. (Note the dependency on t)

Again back to school, a non-linear function is e.g. a parabola
$y=x^2$
Hence, a DE of the form
$x'(t) = x(t)^2$
is non-linear.

Summarizing, every DE which could be written in the form (*) is a linear differential equation. All other equations are non-linear.

Note that linear / non-linear has nothing to do with RK solver. But it is true that linear DEs could mostly easily be solved analytically, and therefore a RK solver is not necessary. But it is not true that DEs which are solved by an RK solver are non-linear.

Check also https://www.physicsforums.com/showthread.php?t=628922

 

[Mute]

If you have a coupled set of differential equations for some set of functions ${y_1(t), y_2(t), \dots, y_n(t)}$, the equations are non-linear if any products of these functions or their derivatives appear.

e.g.,

$$y_1(t)^2,~y_{17}(t)y_2(t),~y_3(t)\dot{y}_4(t),~ \ddot{y}_9(t)y_8(t)y_2^{(400)}(t).$$

It is also non-linear if you have a function of one of the y's, e.g.,

$$\sin(y_{27}(t)),~\exp(-y_{12}(t)^2/2).$$

Note that terms such as

$$ty_5(t),~t^2 + y_2(t)$$

are linear as far as the differential equation is concerned. Linear or non-linear in the context of DEs is referring only to the functions you are solving for, not to the variables the function is a function of.

 

[StationZero]

I guess what I am confused about is the notion of non-linear. I understand that squaring the coefficient defining the slope, or cubing it, etc., will lead to a nonlinear solution there, but there is another definition of non-linear, isn't there? One relating to feedback-dependent effects in the solution to the equations, such as the solutions to the Einstein field equations?

 

[Unstable]

Einstein… okay I am not sure what you mean. This guy made definitely some non-linear stuff.

From your initial post I sought you have some problems to determine if an ordinary differential equation is linear or non-linear.

Before you start with some fancy DEs you have to understand what a DE actually is.

I repeat the simplest example again (this is actually a test for students in an oral examination if they got it or not, and believe me 20-25% will fail…)

Solve

$x'(t) = -k \cdot x(t) \, , \qquad x(0)=x^0$

What is the solution?

If you are not able to write down the solution immediately forget about Einstein...