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Solutions which are singular in the sense that the initial value problem fails to have a unique solution need not be singular functions. In some cases, the term singular solution is used to mean a solution at which there is a failure of uniqueness to the initial value problem at every point on the curve.
In mathematics and logic, the term "uniqueness" refers to the property of being the one and only object satisfying a certain condition. [1] This sort of quantification is known as uniqueness quantification or unique existential quantification, and is often denoted with the symbols "∃!"
The Banach fixed point theorem is then invoked to show that there exists a unique fixed point, which is the solution of the initial value problem. An older proof of the Picard–Lindelöf theorem constructs a sequence of functions which converge to the solution of the integral equation, and thus, the solution of the initial value problem.
The solution is unique if and only if the rank equals the number of variables. Otherwise the general solution has k free parameters where k is the difference between the number of variables and the rank; hence in such a case there are an infinitude of solutions.
Two other solutions are x = 3, y = 6, z = 1, and x = 8, y = 9, z = 2. There is a unique plane in three-dimensional space which passes through the three points with these coordinates, and this plane is the set of all points whose coordinates are solutions of the equation.
The set of all possible solutions is called the solution set. [5] A linear system may behave in any one of three possible ways: The system has infinitely many solutions. The system has a unique solution. The system has no solution.
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The process of row reduction makes use of elementary row operations, and can be divided into two parts.The first part (sometimes called forward elimination) reduces a given system to row echelon form, from which one can tell whether there are no solutions, a unique solution, or infinitely many solutions.