# I What is term for DEQ that only has terms of a derivative?

#### swampwiz

For a DEQ like this:

y = y( x )

a y'''' + b y''' + c y'' + d y' + f y = g( x )

where a, b, c, d, f are constants.

I would think it would be called a "constant coefficient DEQ", but a DEQ like this would also be called this

a y y'' + b ( y' )2 = g( x )

but I am only interested in the term for a DEQ in which every term has a single factor that is a derivative.

If I were naming it, I would call it a "polydifferential" so that it would correspond with the term "polynomial", which of course is what the polydifferential would transform into after presuming the natural exponential function for y( x ).

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#### fresh_42

Mentor
2018 Award
For a DEQ like this:

y = y( x )

a y'''' + b y''' + c y'' + d y' + f y = g( x )

where a, b, c, d, f are constants.

I would think it would be called a "constant coefficient DEQ", but a DEQ like this would also be called this

a y y'' + b ( y' )2 = g( x )

but I am only interested in the term for a DEQ in which every term has a single factor that is a derivative.

If I were naming it, I would call it a "polydifferential" so that it would correspond with the term "polynomial", which of course is what the polydifferential would transform into after presuming the natural exponential function for y( x ).
It is called a (ordinary) linear differential equation.
One can write it as linear equation
$$\begin{bmatrix} a_0(x) , a_1(x) , \ldots , a_n(x) \end{bmatrix} \cdot \begin{bmatrix} y^{(n)}(x) \\ y^{(n-1)}(x) \\ \vdots \\ y^0 (x) \end{bmatrix} = b(x)$$

#### Mark44

Mentor
a y'''' + b y''' + c y'' + d y' + f y = g( x )
where a, b, c, d, f are constants.

I would think it would be called a "constant coefficient DEQ", but a DEQ like this would also be called this
This one is a linear, constant coefficient, nonhomogeneous, fourth-order diff. equation.
Linear because all of the terms involve only the unknown function (y(x)) or its derivatives to the power 1, and because none of the dependent variables (i.e., y, y', y'', etc..) are multiplied together
Constant coefficient because all terms are multiplied only by constants.
Nonhomogeneous because of the g(x) term on the right side. (Moving to the left side doesn't change this.)
Fourth-order because the highest derivative is a fourth derivative.

I would think it would be called a "constant coefficient DEQ", but a DEQ like this would also be called this

a y y'' + b ( y' )2 = g( x )
This one is nonlinear because of the term with yy'' and because of the (y')2 term.
swampwiz said:
but I am only interested in the term for a DEQ in which every term has a single factor that is a derivative.
Your first example meets this requirement if a = b = c = d = 1, so that we could write it as $y^{(4)} + y^{(3)} + y'' + y' + y = g(x)$

#### swampwiz

Your first example meets this requirement if a = b = c = d = 1, so that we could write it as $y^{(4)} + y^{(3)} + y'' + y' + y = g(x)$
I meant to say only one non-constant factor.

#### swampwiz

OK, so it seems that this is called "linear, constant coefficient DEQ". But I like the term "polydifferential".

"What is term for DEQ that only has terms of a derivative?"

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