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In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form + ′ + ″ + () = where a 0 (x), ..., a n (x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y′, ..., y (n) are the successive derivatives of an unknown function y of ...
Likewise, in differential geometry, the differential of a function at a point is a linear function of a tangent vector (an "infinitely small displacement"), which exhibits it as a kind of one-form: the exterior derivative of the function.
Linear differential equations frequently appear as approximations to nonlinear equations. These approximations are only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations.
In mathematics, a fundamental matrix of a system of n homogeneous linear ordinary differential equations ˙ = () is a matrix-valued function () whose columns are linearly independent solutions of the system. [1]
In mathematics (including combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients [1]: ch. 17 [2]: ch. 10 (also known as a linear recurrence relation or linear difference equation) sets equal to 0 a polynomial that is linear in the various iterates of a variable—that is, in the values of the elements of a sequence.
In calculus, the derivative of any linear combination of functions equals the same linear combination of the derivatives of the functions; [1] this property is known as linearity of differentiation, the rule of linearity, [2] or the superposition rule for differentiation. [3]
In mathematics, the Wronskian of n differentiable functions is the determinant formed with the functions and their derivatives up to order n – 1.It was introduced in 1812 by the Polish mathematician Józef Wroński, and is used in the study of differential equations, where it can sometimes show the linear independence of a set of solutions.
In mathematics, a Green's function (sometimes improperly termed a Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if is a linear differential operator, then
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