Definitions for linear differential equation
lin·ear dif·fer·en·tial equa·tion

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1. Linear differential equation

   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 a 0 ( x ) y + a 1 ( x ) y ′ + a 2 ( x ) y ″ ⋯ + a n ( x ) y ( n ) = b ( x ) {\displaystyle a_{0}(x)y+a_{1}(x)y'+a_{2}(x)y''\cdots +a_{n}(x)y^{(n)}=b(x)} where a0(x), ..., an(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 the variable x. Such an equation is an ordinary differential equation (ODE). A linear differential equation may also be a linear partial differential equation (PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are partial derivatives. A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of integrals. This is also true for a linear equation of order one, with non-constant coefficients. An equation of order two or higher with non-constant coefficients cannot, in general, be solved by quadrature. For order two, Kovacic's algorithm allows deciding whether there are solutions in terms of integrals, and computing them if any. The solutions of homogeneous linear differential equations with polynomial coefficients are called holonomic functions. This class of functions is stable under sums, products, differentiation, integration, and contains many usual functions and special functions such as exponential function, logarithm, sine, cosine, inverse trigonometric functions, error function, Bessel functions and hypergeometric functions. Their representation by the defining differential equation and initial conditions allows making algorithmic (on these functions) most operations of calculus, such as computation of antiderivatives, limits, asymptotic expansion, and numerical evaluation to any precision, with a certified error bound.

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1. Linear differential equation

   In mathematics, linear differential equations are differential equations that have solutions that can be added together to form other solutions. They can be ordinary or partial. The solutions to linear equations form a vector space.

How to pronounce linear differential equation?

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Numerology

1. Chaldean Numerology

   The numerical value of linear differential equation in Chaldean Numerology is: 4

2. Pythagorean Numerology

   The numerical value of linear differential equation in Pythagorean Numerology is: 9

References

1. ^ Wikipedia
   https://en.wikipedia.org/wiki/Linear_Differential_Equation
2. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=linear differential equation