Equation

Under an equation is understood in mathematics, a statement about the equality of two terms, with the help of the equal sign ( "=" ) is symbolized. Formally, has an equation of the form

Wherein the term, the left side and the term is called the right side of the equation. Equations are either true or satisfied ( for example ) or false (for example). If at least one of the terms of variables depends, there is only one form of expression; whether the equation is true or false, depends on the specific values ​​used. The values ​​of the variables for which the equation is satisfied, the hot solutions to the equation. If two or more equations specified, one also speaks of a system of equations, a solution must meet all of the same equations simultaneously.

• 3.1 Systems of Linear Equations
• 3.2 Nonlinear Systems of Equations

• 4.1 Analytical solution
• 4.2 Numerical Solution
• 4.3 Qualitative Analysis

Types of equations

Equations are used in many contexts; Accordingly, there are different ways to classify the equations according to various criteria. The respective divisions are to a large extent independently of each other, an equation can fall into several of these groups. Example, it is useful to speak of a system of linear partial differential equations.

View the validity

Identity equations

Equations can be universal, ie, by substituting all variables values ​​from a given basic set, or at least from a pre- defined subset of them be true. The general validity can either be proved by other axioms or are themselves assumed as an axiom.

Examples are:

• The Pythagorean theorem: is true for right triangles, if the right angle opposite side ( hypotenuse ) denotes
• The associative law: is true for all natural numbers; true for all groups ( as an axiom )
• The first binomial formula is true for all real numbers
• Euler's identity is true for all real

In this context one also speaks of a mathematical theorem or law. To distinguish them from non- universal equations and the Kongruenzzeichen ( "≡ " ) is used in identities instead of the equal sign.

Determining equations

Is not generally valid for one equation, there are certain values ​​from the basic amount for which the equation yields a true statement, and certain values ​​for which the equation provides a false statement. Often a task is to determine all the elements of the basic amount for which the equation yields a true statement. This process is called solving the equation. To distinguish them from identity equations such equations are referred to as conditional equations. The amount of the values ​​of the variables for which the equation is true, is called a set of solutions of the equation. If it is in the solution set to the empty set, so the equation is referred to as intractable or impractical.

Whether or not an equation is solved, will depend on the considered base quantity, for example, applies:

• The equation is unsolvable as equation via the natural or rational numbers and has the solution set as an equation over the reals
• The equation is unsolvable as equation over the real numbers and has the solution set as an equation over the complex numbers

In determining equations are sometimes susceptible to variables that are not wanted, but are assumed to be known. Such variables are called parameters. For example, the solution formula for the quadratic equation

Sought in unknown and given parameters and

Substituting one of the two solutions into the equation, then the equation is transformed into an identity that is so for any choice of and true statement. For here the solutions are real, otherwise complex.

Defining equations

Equations may 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 often by the definition characters (": =" ) replaced or written about the equal sign "def".

For example, the derivative of a function at a point is

Defined. Unlike identities definitions are not statements; they are neither true nor false, but only more or less appropriate.

Classification according to right side

Homogeneous equations

A conditional equation of the form

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

Fixed-point equations

A conditional equation of the form

Is called fixed point equation and its solution is called fixed point of the equation. More details about the solutions of such equations say from fixed-point theorems.

Eigenvalue problems

A conditional equation of the form

Is called an eigenvalue problem, where the constant ( the eigenvalue ) and the unknown ( the eigenvector ) to be sought together. Eigenvalue problems have numerous applications in linear algebra, for example, in the analysis and decomposition of matrices, and in application areas, such as structural mechanics and quantum mechanics.

Classification by Linearity

Linear Equations

An equation is called linear if it in the form

Can be brought, where the term is independent of, and the term is linear in, ie

Applies to coefficients. Usefully, the appropriate operations must be defined, so it is necessary that and are made of a vector space, and the solution of the same or a different vector space is searched.

Linear equations are usually much easier to solve than nonlinear. So true of linear equations, the superposition principle: 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 the linearity is at least one solution of a homogeneous equation. Has a homogeneous equation thus has a unique solution, as well as a corresponding inhomogeneous equation has at most one solution. A related, but much more profound statement in the functional analysis is the Fredholm alternative.

Non-linear equations are often distinguished by the type of nonlinearity. Especially in school mathematics, the following basic types are treated by non-linear equations.

Algebraic equations

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

Or cubic equations of the form

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

Fraction equations

Contains an equation a fraction term in which the unknown occurs, at least in the denominator, it is called a fracture equation, for example

By multiplying by the common denominator, in this example, to break equations can be reduced to algebraic equations. Such a multiplication is usually no equivalence transformation and it must be made a case distinction, in the example is not included in the definition area of ​​the fracture equation.

Root equations

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

Root equations are special power equations with exponents. Root equations can be solved by a root is isolated and then the equation with the exponents of the root ( in the example ) is potentiated. This procedure is repeated until all roots have been eliminated. Exponentiation with an even-numbered exponents does not constitute Äquivalenzumforung and therefore is to be made in these cases in determining the solution an appropriate case distinction. In the example, squaring leads to the quadratic equation, the negative solution does not lie in the domain of the initial equation.

Exponential equations

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

Exponential equations can be solved by taking logarithms. Conversely Logarithmusgleichungen - ie equations in which the unknowns than number ( argument of a logarithm function ) occurs - by Exponenzieren solvable.

Trigonometric equations

If the unknowns as an argument at least one angular function, it is called a trigonometric equation, for example,

The solutions of trigonometric equations are periodically repeated in general, provided that the solution set is not as restricted to a certain interval. Alternatively, the solutions can be parameterized by an integer variable. For example, the solutions to the above equation are given as

Classification by this unknown

Algebraic equations

To distinguish equations, where is a real number or a real vector is sought by equations, for example where a function is searched, sometimes the term is used algebraic equation, which term, however, is then not limited to polynomials. This manner of speech is controversial.

Diophantine equations

One examines integer solutions of a scalar equation with integer coefficients, then one speaks of a Diophantine equation. An example of a cubic Diophantine equation

Sought by the integer that satisfy the equation, the numbers here.

Difference equations

Is the unknown a consequence, it is called a differential equation. A known example of a linear second order difference equation

Whose solution for start values ​​and the Fibonacci sequence.

Functional equations

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

Whose solutions are the exponential straight.

Differential equations

If in the equation is a function sought, which occurs with derivatives, it is called a differential equation. Differential equations arise in the modeling of scientific problems are very common. The highest occurring derivative is here called order of the differential equation. We distinguish:

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

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

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

• Stochastic differential equations, which occur in addition to deterministic and stochastic derivative terms, such as the Black-Scholes equation of mathematical finance to model security prices

Integral equations

If the unknown function in an integral, so we speak of an integral equation. Is an example of a linear integral equation 1 Type

If there are several in a row equals sign, then one speaks of a chain of equations. In a chain of equations all separated by equal sign expressions by value should be the same. In this case, each of these expressions is to be considered separately. For example, the equation chain

Wrong because it is broken down into individual equations leads to false statements. It is true, however, for example

Equation chains are interpreted meaningfully in particular because of the transitivity of the equality relation. Equation chains often also occur together with inequalities in assessments, as is true, for example,

Systems of equations

Often several equations that must be satisfied simultaneously considered while looking for several unknowns simultaneously.

Systems of linear equations

An equation system - that is a set of equations - is a linear system of equations, if all of the equations are linear. For example, is

A linear equation system consisting of two equations and three unknowns and. Summing up both the equations and the unknowns to tuples together, so can be a system of equations also be regarded as a single equation for an unknown vector. How to write in linear algebra a system of equations as a vector equation

With a matrix, the unknown vector and the right-hand side, the matrix-vector product. In the above example are

Nonlinear systems of equations

Equation systems whose equations are not all linear, are called non-linear systems of equations. For example, is

A nonlinear system of equations in the unknowns and. For such equations there is no general solution strategies. Often you only have the opportunity approximate solutions with the help of numerical methods to determine. A powerful method of approximation, for example, the Newton method.

A rule of thumb is that the same number of equations as unknowns are needed so that a system of equations has a unique solution. But this is actually only a rule of thumb, to a certain degree it is true because of the main implicit function theorem for real equations with real unknowns.

Solving equations

Analytical solution

Insofar as possible, one tries to determine the solutions of a conditional equation exactly. The most important help in this process are equivalence transformations through which an equation gradually in other equivalent equations ( ie the same amount of solution have ) is reshaped to obtain an equation whose solution can be easily determined.

Numerical Solution

Many equations, especially from scientific applications that can not be solved analytically. In this case, one tries to compute an approximate numerical solution to the computer. Such methods are discussed in numerical mathematics. Many non-linear equations can be solved approximatively by the nonlinearities occurring in the equation can be approximated linearly, and then solved the resulting linear problems ( for example, in Newton's method ). For other classes of problems, such as solving equations in infinite- dimensional spaces, the solution is sought in suitably chosen finite-dimensional subspaces (eg, in the Galerkin method).

Qualitative Analysis

Even if an equation can not be solved analytically, it is still often possible to make statements about the mathematical solution. Particular interest issues if a solution exists at all, whether it is unique, and whether it depends continuously on the parameters of the equation. If this is the case, one speaks of a problem correctly identified. A qualitative analysis is important also, or just in the numerical solution of an equation in order to ensure that the numerical solution actually delivers an approximate solution of the equation.