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A homogeneous linear differential equation of the second order may be written ″ + ′ + =, and its characteristic polynomial is + +. If a and b are real , there are three cases for the solutions, depending on the discriminant D = a 2 − 4 b .
A linear differential equation is homogeneous if it is a homogeneous linear equation in the unknown function and its derivatives. It follows that, if φ(x) is a solution, so is cφ(x), for any (non-zero) constant c. In order for this condition to hold, each nonzero term of the linear differential equation must depend on the unknown function or ...
If u(x) and v(x) are two non-trivial continuous linearly independent solutions to a homogeneous second order linear differential equation with x 0 and x 1 being successive roots of u(x), then v(x) has exactly one root in the open interval (x 0, x 1). It is a special case of the Sturm-Picone comparison theorem.
Consider the general, homogeneous, second-order linear constant coefficient ordinary differential equation. (ODE) ″ + ′ + =, where ,, are real non-zero coefficients. . Two linearly independent solutions for this ODE can be straightforwardly found using characteristic equations except for the case when the discriminant, , vanish
The differential equation is said to be in Sturm–Liouville form or self-adjoint form.All second-order linear homogenous ordinary differential equations can be recast in the form on the left-hand side of by multiplying both sides of the equation by an appropriate integrating factor (although the same is not true of second-order partial differential equations, or if y is a vector).
The order of the differential equation is the highest order of derivative of the unknown function that appears in the differential equation. For example, an equation containing only first-order derivatives is a first-order differential equation, an equation containing the second-order derivative is a second-order differential equation, and so on.
In mathematics, Abel's identity (also called Abel's formula [1] or Abel's differential equation identity) is an equation that expresses the Wronskian of two solutions of a homogeneous second-order linear ordinary differential equation in terms of a coefficient of the original differential equation.
[3] [4] The characteristic equation can only be formed when the differential or difference equation is linear and homogeneous, and has constant coefficients. [1] Such a differential equation, with y as the dependent variable, superscript (n) denoting n th-derivative, and a n, a n − 1, ..., a 1, a 0 as constants,
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