What are constant coefficients in ODEs?

[autodidude]

It says on Wikipedia in the article on differential equations that: 'a differential equation is linear if the unknown function and its derivatives appear to the power 1 (products are not allowed) and nonlinear otherwise'

Are these products between any of the variables that appear? So, are products between derivatives, the unknown function AND the independent variable not allowed or is it just between the derivatives and unknown functions?

In a book on ODEs I found in the library, it says:

..the general form linear ordinary differential equation of order n is
$$a_0(x)y^{(n)}+a_1(x)y^{(n-1)}+...}a_n(x)y=g(x)$$

$$a_n(x)$$ are coefficients which are functions of x...so can these be any random function of x that doesn't really relate to the unknown function we're trying to find? Are they usually constants?

And what is the g(x) on the RHS?

Thanks

 

[Astronuc]

See this page for some additional discussion on terminology -
http://hyperphysics.phy-astr.gsu.edu/hbase/diff.html#c6
http://hyperphysics.phy-astr.gsu.edu/hbase/diff.html
http://hyperphysics.phy-astr.gsu.edu/hbase/de.html#deh
http://hyperphysics.phy-astr.gsu.edu/hbase/math/spfun.html#c1

Linear means that the derivatives are of order one, e.g., y''', y'', y' or the function y, as cited in your example. There are not powers > 1, e.g., (y')ⁿ, and no products, e.g., y y' or y' y''.

The coefficients are functions of the independent variable, e.g., a(x) = x, or 1/x, or x², or they can be constants.

In one's example, g(x) is a source term or forcing function (which makes more sense if x = t), and y(x) would be expressed as some function of g(x) and the coefficients of the derivatives of yⁱ and y.

Here is a good reference on differential equations - http://tutorial.math.lamar.edu/Classes/DE/DE.aspx

 

[autodidude]

I'm not sure what you mean by forcing function...but if we had something like

y''+sin(x+y)=sin(x)

Would sin(x) be the g(x)? If we had known additional functions of x that aren't coefficients of the unknown function or the derivatives, then they would be part of the g(x) too?

 

[JJacquelin]

y''+sin(x+y)=sin(x) is NOT a linear ODE because sin(x+y) is not a linear function of y.
y''+sin(x+y)=0 is NOT a linear ODE because sin(x+y) is not a linear function of y.
y''+sin(y)=sin(x) is NOT a linear ODE because sin(y) is not a linear function of y.
y''+sin(y)=0 is NOT a linear ODE because sin(y) is not a linear function of y.

y''+sin(x)*y=sin(x) is a linear ODE because sin(x)*y is a linear function of y
.

 

[Astronuc]

I'm not sure what you mean by forcing function...but if we had something like

y''+sin(x+y)=sin(x)

Would sin(x) be the g(x)? If we had known additional functions of x that aren't coefficients of the unknown function or the derivatives, then they would be part of the g(x) too?

The nonlinearity was addressed in the previous post.

Besides linearity, one has homogenous and non-homogenous differential equations.

y'' + P(x) y' + Q(x) y = R(x) is nonhomogenous, if R(x) ≠ 0, and

y'' + P(x) y' + Q(x) y = 0 is a homogenous differential equation.

There are many special types of ODEs: http://mathworld.wolfram.com/OrdinaryDifferentialEquation.html

http://mathworld.wolfram.com/Second-OrderOrdinaryDifferentialEquationSecondSolution.html

ODEs with constant coefficients are fairly simple.