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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.
A differential equation can be homogeneous in either of two respects.. A first order differential equation is said to be homogeneous if it may be written (,) = (,),where f and g are homogeneous functions of the same degree of x and y. [1]
If a second-order differential equation has a characteristic equation with complex conjugate roots of the form r 1 = a + bi and r 2 = a − bi, then the general solution is accordingly y(x) = c 1 e (a + bi )x + c 2 e (a − bi )x. By Euler's formula, which states that e iθ = cos θ + i sin θ, this solution can be rewritten as follows:
Sturm–Liouville theory is a theory of a special type of second-order linear ordinary differential equation. Their solutions are based on eigenvalues and corresponding eigenfunctions of linear operators defined via second-order homogeneous linear equations.
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 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.
First-order means that only the first derivative of y appears in the equation, and higher derivatives are absent. Without loss of generality to higher-order systems, we restrict ourselves to first-order differential equations, because a higher-order ODE can be converted into a larger system of first-order equations by introducing extra variables.
The following assumptions are made while deriving Euler's formula: [3] The material of the column is homogeneous and isotropic. The compressive load on the column is axial only. The column is free from initial stress. The weight of the column is neglected. The column is initially straight (no eccentricity of the axial load).