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Order Equation Applications Airy: 2 ... List of linear ordinary differential equations. Add languages ...
The highest order of derivation that appears in a (linear) differential equation is the order of the equation. The term b(x), which does not depend on the unknown function and its derivatives, is sometimes called the constant term of the equation (by analogy with algebraic equations), even when this term is a non-constant function.
This is an outline of topics related to linear algebra, the branch of mathematics concerning linear equations and linear maps and their representations in vector spaces and through matrices. Linear equations
Just as with any linear system of two equations, two solutions may be called linearly-independent if + = implies = =, or equivalently that | ˙ ˙ | is non-zero. This notion is extended to second-order systems, and any two solutions to a second-order ODE are called linearly-independent if they are linearly-independent in this sense.
0th-order contact if the curves have a simple crossing (not tangent). 1st-order contact if the two curves are tangent. 2nd-order contact if the curvatures of the curves are equal. Such curves are said to be osculating. 3rd-order contact if the derivatives of the curvature are equal. 4th-order contact if the second derivatives of the curvature ...
When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set. For linear equations, logical independence is the same as linear independence. The equations x − 2y = −1, 3x + 5y = 8, and 4x + 3y = 7 are linearly dependent. For example ...
Conversely, every line is the set of all solutions of a linear equation. The phrase "linear equation" takes its origin in this correspondence between lines and equations: a linear equation in two variables is an equation whose solutions form a line. If b ≠ 0, the line is the graph of the function of x that has been defined in the preceding ...
In three-dimensional Euclidean space, these three planes represent solutions to linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations. Linear algebra is the branch of mathematics concerning linear equations such as:
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