Differential algebra and differential-algebraic equations Could you give me some information about differential algebra? What is it about? 

Differential-algebraic equations (DAEs) are polynomials with complex coefficients and the unknown variables are $z, x, x'$. 

Is this correct? 

What is the difference between them and the ODEs? 

Two possible solutions of DAEs $C^{\infty}$ functions and formal power series, right? 

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EDIT: 

About differential algebra I found also this link. 

Does this mean that Differential algebra is about differential equations over a field or a ring?
These are two distinct and not very related concepts (though not completely unrelated).

Differential algebra is the study of differential rings and fields and related structures. Let me briefly mention some things about differential fields to give you some idea about what differential algebra is about.

A differential field is a field $\mathbb F$ together with a function $\partial : \mathbb F \to \mathbb F$, called a derivation, that satisfies the product rule:
$$\partial(xy) = \partial(x)y + x\partial(y)$$
An element $x \in \mathbb F$ is called a constant of the differential field if
$$\partial(x) = 0.$$
The set of all constants form a subfield of the differential field.

An example of a differential ring is $\mathbb R(t)$, the field of rational functions in $t$ over $\mathbb R$, with the derivation $\frac{d}{dt}$, differentiation with respect to $t$. The constants of $\mathbb R(t)$ is $\mathbb R$.

Elements of differential algebra are used in e.g. differential Galois theory and symbolic integration.

A differential algebraic system of equations is a system of equations where some equations are algebraic equations and some are differential equations. The equations need not be polynomial. I say system of equations, because if it is not a system of equations, i.e. there is only one equation, it will either be purely algebraic or differential.

An example of a DAE system is the equations describing the motions of a planar pendulum, having position $(x,y)$, velocity $(u,v)$, all functions of time $t$, with length $L$:
$$\begin{align}
\dot x &amp;= u \\
\dot y &amp;= v \\
\dot u &amp;= \lambda x \\
\dot v &amp;= \lambda y - g \\
L^2 &amp;= x^2 + y^2
\end{align}$$
as you can see, the last equation is algebraic and not differential.

I have explained the difference between an ODE and a DAE here: What is the difference between an implicit ordinary differential equation and a differential algebraic equation?

As an aside, your description of DAEs ("polynomials with complex coefficients and the unknown variables are $z,x,x'$") got me thinking of holonomic functions, and while they are not exactly what you described, they do come close. A holonomic function $y(t)$ is a function that satisfies
$$a_r(t) y^{(r)}(t) + a_{r-1}(t) y^{(r-1)}(t) + \dots a_1(t)y'(t) + a_0(t)y(t) = 0$$
where each $a_i(t)$ is a polynomial in $t$.
as a PhD of DAE I want you to read these 3 books in this order:


Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations (Classics in Applied Mathematics) 2nd Edition
by K. E. Brenan (Author), S. L. Campbell (Author), L. R. Petzold (Author)


Singular Systems of Differential Equations Vol. I and II, 
by S. L. Campbell (Author)