Linear and non linear differential

[killerfish]

Hi guys,

how do i tell a differential equation is a linear or non-linear? i have hard time understanding from my textbook/google, i need some examples to understand... I am studying separtion,exact diff, integrating factor, i have no idea what is going on... any kind soul can help me...

Thanks you..

 

[tiny-tim]

fishy welcome to pf!

hi killerfish! fishy welcome to pf!

how do i tell a differential equation is a linear or non-linear?

A linear differential equation is of the form ∑a_nDⁿ = 0.

For details, see http://en.wikipedia.org/wiki/Linear_differential_equation

 

[n1person]

generally speaking if
a) a derivative is raised to a power
b) there is something of the form f(x)g(y)y^(n)(x) (y^(n)(x) is the nth derivative of y)

then the equation is non-linear

 

[HallsofIvy]

Do you understand what a "non-linear" function is? A linear function involves only addition, subtraction, and multiplication or division by a constant. Any other function is non-linear. A differential equation is "non-linear" if it involves any non-linear functions of the dependent variable. For example, y'= 3y²+ x is non-linear because of the square of the dependent variable, y. y'= 3y+ x² is not non-linear because it is the independent variable, x, that is squared.

 

[n1person]

edit: nevermind

 

[ravioli]

Another way to look at it is the following:

A function is linear if f(a_1 \bold{x_1}+a_2 \bold{x_2})=a_1 f(\bold{x_1})+a_2 f(\bold{x_2})} With a_1, a_2 being scalars, and x_1, x_2 being variables. That is the definition of linearity, and if you apply it to what HallsofIvy said below, you can see why squaring of the dependent variable is nonlinear (there's going to be an extra cross-term product).

Basically if a function is linear then if you have two inputs, those two inputs will act independently of each other. So let's say we want an example of how this property can be useful in a non-mathematical way. Let's say you have two sine waves going into an amplifier with two distinct frequencies f_1, f_2. We want the amplifier to amplify those two frequencies separately, without any impact of one frequency on the other. So basically we want the amplifier to be linear.

 

[killerfish]

thanks guys for showing out the points I am much clearer now...=)