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The zeros of two linearly independent solutions of the Airy equation ″ = alternate, as predicted by the Sturm separation theorem.. In mathematics, in the field of ordinary differential equations, Sturm separation theorem, named after Jacques Charles François Sturm, describes the location of roots of solutions of homogeneous second order linear differential equations.
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.
Reduction of order (or d’Alembert reduction) is a technique in mathematics for solving second-order linear ordinary differential equations. It is employed when one solution y 1 ( x ) {\displaystyle y_{1}(x)} is known and a second linearly independent solution y 2 ( x ) {\displaystyle y_{2}(x)} is desired.
Their solutions are based on eigenvalues and corresponding eigenfunctions of linear operators defined via second-order homogeneous linear equations. The problems are identified as Sturm–Liouville problems (SLP) and are named after J. C. F. Sturm and J. Liouville , who studied them in the mid-1800s.
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:
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).
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 .
Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly.