- #1

StationZero

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- Thread starter StationZero
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- #1

StationZero

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- #2

Unstable

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[itex] y= a\cdot x + b [/itex]

Now a differential equation of the form

[itex] x'(t) = A(t) \cdot x(t) + b(t) \qquad \qquad (*) [/itex]

is a linear differential equation. (Note the dependency on t)

Again back to school, a non-linear function is e.g. a parabola

[itex] y=x^2 [/itex]

Hence, a DE of the form

[itex] x'(t) = x(t)^2 [/itex]

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

- #3

Mute

Homework Helper

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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.

- #4

StationZero

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- #5

Unstable

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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

[itex] x'(t) = -k \cdot x(t) \, , \qquad x(0)=x^0 [/itex]

What is the solution?

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

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

[itex] x'(t) = -k \cdot x(t) \, , \qquad x(0)=x^0 [/itex]

What is the solution?

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

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