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A variable may denote an unknown number that has to be determined; in which case, it is called an unknown; for example, in the quadratic equation ax 2 + bx + c = 0, the variables a, b, c are parameters, and x is the unknown.
For example, the equation x + y = 2x – 1 is solved for the unknown x by the expression x = y + 1, because substituting y + 1 for x in the equation results in (y + 1) + y = 2(y + 1) – 1, a true statement. It is also possible to take the variable y to be the unknown, and then the equation is solved by y = x – 1.
A difference equation is an equation where the unknown is a function f that occurs in the equation through f(x), f(x−1), ..., f(x−k), for some whole integer k called the order of the equation. If x is restricted to be an integer, a difference equation is the same as a recurrence relation
In mathematics, a quadratic equation (from Latin quadratus 'square') is an equation that can be rearranged in standard form as [1] + + =, where the variable x represents an unknown number, and a, b, and c represent known numbers, where a ≠ 0. (If a = 0 and b ≠ 0 then the equation is linear, not quadratic.)
g(x) is a constant, a polynomial function, exponential function , sine or cosine functions or , or finite sums and products of these functions (, constants). The method consists of finding the general homogeneous solution y c {\displaystyle y_{c}} for the complementary linear homogeneous differential equation
Consider a system of n linear equations for n unknowns, represented in matrix multiplication form as follows: = where the n × n matrix A has a nonzero determinant, and the vector = (, …,) is the column vector of the variables. Then the theorem states that in this case the system has a unique solution, whose individual values for the unknowns ...
The system + =, + = has exactly one solution: x = 1, y = 2 The nonlinear system + =, + = has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while + + =, + + =, + + = has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be ...
Let y (n) (x) be the nth derivative of the unknown function y(x).Then a Cauchy–Euler equation of order n has the form () + () + + =. The substitution = (that is, = (); for <, in which one might replace all instances of by | |, extending the solution's domain to {}) can be used to reduce this equation to a linear differential equation with constant coefficients.