Equation

In mathematics, an equation is a statement about the equality of two terms, symbolized by the equals sign ("="). Formally, an equation has the form

$T_{1}=T_{2}$ ,

where the term $T_{1}$ is called the left side and the term $T_{2}$ is called the right side of the equation. Equations are either true or satisfied (for example, 1 = 1 ) or false (for example, 1 = 2 ). If at least one of the terms T_1, T_2 depends on variables, only one propositional form is present; whether the equation is true or false then depends on the specific values used. The values of the variables for which the equation is satisfied are called solutions of the equation. If two or more equations are given, one also speaks of a system of equations; a solution of the same must satisfy all equations simultaneously.

Oldest printed equation (1557), in today's notation "14x + 15 = 71".

Equation types

Equations are used in many contexts; accordingly, there are various ways to classify equations according to different viewpoints. To a large extent, the respective classifications are independent of each other; an equation may fall into several of these groups. For example, it is useful to speak of a system of linear partial differential equations.

Classification according to validity

Identity equations

→ Main article: Identity equation

Equations can be generally valid, i.e. they can be true by inserting all variable values from a given basic set or at least from a previously defined subset of it. The generality can either be proved with other axioms or be assumed as an axiom itself.

Examples include:

the Pythagorean theorem: $a^{2}+b^{2}=c^{2}$ is true for right triangles, if denotes the side opposite to the right angle (hypotenuse) and denote the cathets
the associative law: is true for all natural numbers and in general for any elements a group (as an axiom).
the first binomial formula: is true for all real numbers
the Eulerian identity: $e^{i \varphi} = \cos\left(\varphi \right) + i \sin\left( \varphi\right)$ is true for all real φ $\varphi$

In this context, one also speaks of a mathematical theorem or law. To distinguish from equations that are not generally valid, the congruence sign ("≡") is also used for identities instead of the equals sign.

Equations of determination

Often, a task consists of determining all variable assignments for which the equation becomes true. This process is called solving the equation. To distinguish them from identity equations, such equations are called determination equations. The set of variable assignments for which the equation is true is called the solution set of the equation. If the solution set is the empty set, the equation is called unsolvable or unsatisfiable.

Whether an equation is solvable or not may depend on the basic set under consideration, for example holds:

the equation is unsolvable as an equation over the natural or the rational numbers and has the solution set $\lbrace \sqrt{2}, - \sqrt{2} \rbrace$ as an equation over the real numbers
the equation is unsolvable as an equation over the real numbers and has the solution set $\lbrace \sqrt{2}i, -\sqrt{2}i \rbrace$ as an equation over the complex numbers

In equations of determination, variables sometimes occur that are not searched for, but are assumed to be known. Such variables are called parameters. For example, the solution formula for the quadratic equation is

$x^2+px+q \; = \; 0$

with searched unknown and given parameters and

$x_{1,2} \; = \; -\frac{p}{2}\pm \sqrt{\frac{p^2}{4}-q}$ .

If one substitutes one of the two solutions $x_{1},x_{2}$ into the equation, the equation turns into an identity, thus becomes a true statement for any choice of and . For $4q \leq p^2$ the solutions here are real, otherwise complex.

Definition equations

Equations can also be used to define a new symbol. In this case, the symbol to be defined is written on the left, and the equal sign is often replaced by the definition sign (":=") or written "def" above the equal sign.

For example, the derivative of a function at a point is $x_{0}$ given by

$f'(x_0) := \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0}$

defined. Unlike identities, definitions are not statements; thus they are neither true nor false, but only more or less expedient.

Classification according to right side

Homogeneous equations

A determinant equation of the form

T(x) = 0

is called a homogeneous equation. If is a function, the solution also called the zero of the function. Homogeneous equations play an important role in the solution structure of systems of linear equations and linear differential equations. If the right side of an equation is nonzero, the equation is called inhomogeneous.

Fixed point equations

→ Main article: Fixed point (mathematics)

A determinant equation of the form

T(x) = x

is called fixed point equation and its solution is called fixed point of the equation. More details about the solutions of such equations are given by fixed point theorems.

Eigenvalue problems

→ Main article: Eigenvalue problem

A determinant equation of the form

$T(x) = \lambda x$

is called an eigenvalue problem, where the constant λ $\lambda$ (the eigenvalue) and the unknown $x\neq 0$ (the eigenvector) are searched for together. Eigenvalue problems have many applications in linear algebra, for example in the analysis and decomposition of matrices, and in application areas, for example in structural mechanics and quantum mechanics.

Classification according to linearity

Linear equations

→ Main article: Linear equation

An equation is called linear if it takes the form

$T\left(x\right) = a$

where the term is independent of and the term is T(x) linear in , i.e.

$T\left(\lambda x + \mu y\right) = \lambda T\left( x \right) + \mu T\left( y\right)$

$\lambda, \mu$ holds for coefficients λ Sensibly, the matching operations must be defined, so it is necessary that T(x) and are from a vector space , and the solution is sought from the same or another vector space

Linear equations are usually much easier to solve than nonlinear ones. Thus, the superposition principle applies to linear equations: The general solution of an inhomogeneous equation is the sum of a particular solution of the inhomogeneous equation and the general solution of the associated homogeneous equation.

Because of linearity, at least x=0 is a solution of a homogeneous equation. Thus, if a homogeneous equation has a unique solution, a corresponding inhomogeneous equation also has at most one solution. A related but much more profound statement in functional analysis is the Fredholm alternative.

Nonlinear equations

Nonlinear equations are often distinguished according to the type of nonlinearity. In particular, in school mathematics, the following basic types of nonlinear equations are treated.

Algebraic equations

→ Main article: Algebraic equation

If the term of the equation is a polynomial, it is called an algebraic equation. If the polynomial is at least of degree two, the equation is called nonlinear. Examples are general quadratic equations of the form

$ax^{2}+bx+c=0$

or cubic equations of the form

ax^3 + bx^2 + cx + d = 0 .

For polynomial equations up to degree four there are general solution formulas.

Fraction equations

→ Main article: Fractional equation

If an equation contains a fraction term where the unknown occurs at least in the denominator, it is called a fraction equation, for example

$\frac{x+2}{x^2+3} = \frac{2}{x+1}$ .

By multiplying by the major denominator, in the example (x^2+3)(x+1) , fractional equations can be reduced to algebraic equations. Such a multiplication is usually not an equivalence transformation and a case distinction must be made; in the example, x=-1 is not included in the domain of definition of the fraction equation.

Root equations

→ Main article: Root equation

In root equations, the unknown is at least once under a root, for example

$\sqrt{x} = 1-x$

Root equations are special power equations with exponent $\tfrac1n$ . Root equations can be solved by isolating a root and then solving the equation with the root exponent (in the example, n=2 ) is exponentiated. This procedure is repeated until all roots are eliminated. Exponentiating with even exponents does not represent an equivalence transformation, and so in these cases an appropriate case distinction must be made when determining the solution. In the example, squaring leads to the quadratic equation x = (1-x)^2 whose negative solution does not lie in the domain of definition of the initial equation.

Exponential equations

In exponential equations, the unknown is in the exponent at least once, for example:

$2^{3x+2} = 4^{x+1}$

Exponential equations can be solved by logarithmizing. Conversely, logarithmic equations - i.e. equations in which the unknown occurs as a numerus (argument of a logarithmic function) - can be solved by exponentiation.

Trigonometric equations

→ Main article: Trigonometric equation

If the unknowns occur as arguments of at least one angular function, the equation is called a trigonometric equation, for example

$\sin(x) = \cos(x)$

The solutions of trigonometric equations generally repeat periodically unless the solution set is restricted to a particular interval, such as $[0,2\pi )$ , is restricted. Alternatively, the solutions can be parameterized by an integer variable For example, the solutions of the above equation are given as

with $x = \frac{\pi}{4} + \pi k$ $k \in \mathbb{Z}$ .

Classification according to searched unknowns

Algebraic equations

→ Main article: Algebraic equation

In order to distinguish equations in which a real number or a real vector is sought from equations in which, for example, a function is sought, the term algebraic equation is sometimes also used, although this term is then not restricted to polynomials. However, this way of speaking is controversial.

Diophantine equations

→ Main article: Diophantine equation

If you are looking for integer solutions to a scalar equation with integer coefficients, it is called a Diophantine equation. An example of a cubic Diophantine equation is

2x^3 - x^2 - 8x = -4 ,

of the integer satisfying the equation are sought, here the numbers $x \in \mathbb{Z}$ $x=\pm 2$ .

Difference equations

→ Main article: Difference equation

If the unknown is a sequence, it is called a difference equation. A well known example of a second order linear difference equation is

$x_n - x_{n-1} - x_{n-2} = 0$ ,

whose solution for initial values $x_{0}=0$ and x_1 = 1 $1, 2, 3, 5, 8, 13, \ldots$ is the Fibonacci sequence

Functional equations

→ Main article: Functional equation

If the unknown of the equation is a function that occurs without derivatives, it is called a functional equation. An example of a functional equation is

f(x+y) = f(x)f(y) ,

whose solutions are just the exponential functions . $f(x)=a^{x}$

Differential equations

→ Main article: Differential equation

If a function is sought in the equation that occurs with derivatives, it is called a differential equation. Differential equations occur very often in the modeling of scientific problems. The highest occurring derivative is called order of the differential equation. A distinction is made between:

ordinary differential equations where only derivatives with respect to a single variable occur, for example the linear ordinary differential equation of the first order

f'(x) + x f(x) = 0

partial differential equations in which partial derivatives occur after several variables, for example the linear transport equation of first order

$\frac{\partial f(x,t)}{\partial t} + \frac{\partial f(x,t)}{\partial x} = 0$

differential-algebraic equations in which both algebraic equations and differential equations occur together, for example the Euler-Lagrange equations for a mathematical pendulum

$\begin{align} \ddot{x}_1 & = 2 x_1 \lambda \\ \ddot{x}_2 & = 2 x_2 \lambda - 1 \\ 0 & = x_1^2 + x_2^2 - 1 \end{align}$

Stochastic differential equations in which stochastic derivative terms occur in addition to deterministic ones, e.g. the Black-Scholes equation in financial mathematics for modeling security prices.

${\rm d}S_t = r S_t {\rm d}t + \sigma S_t {\rm d}W_t$

Integral equations

→ Main article: Integral equation

If the function you are looking for occurs in an integral, it is called an integral equation. An example of a linear integral equation of the 1st kind is

$\int _{0}^{x}(x-t)f(t)~\mathrm {d} t=x^{3}$ .

Chains of equations

If there are several equal signs in one line, this is called a chain of equations. In a chain of equations, all expressions separated by an equal sign should have the same value. Each of these expressions is to be considered separately. For example, the equation chain

17+3 = 20/2 = 10+7 = 17

is wrong, because it leads to wrong statements when divided into single equations. True on the other hand is for example

17+3 = 40/2 = 10+10 = 20 .

Chains of equations can be interpreted in a meaningful way, in particular because of the transitivity of the equality relation. Chains of equations often also appear together with inequalities in estimates, for example, for $n\ge 3$

$2n^2 = n^2+n^2 \ge n^2+3n > n^2+2n+1 = (n+1)^2$ .

Equation

Equation types

Classification according to validity

Identity equations

Equations of determination

Definition equations

Classification according to right side

Homogeneous equations

Fixed point equations

Eigenvalue problems

Classification according to linearity

Linear equations

Nonlinear equations

Algebraic equations

Fraction equations

Root equations

Exponential equations

Trigonometric equations

Classification according to searched unknowns

Algebraic equations

Diophantine equations

Difference equations

Functional equations

Differential equations

Integral equations

Chains of equations

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