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In mathematics, the Hill equation or Hill differential equation is the second-order linear ordinary differential equation. where is a periodic function with minimal period and average zero. By these we mean that for all. and. and if is a number with , the equation must fail for some . [1] It is named after George William Hill, who introduced it ...
Sturm–Liouville theory. In mathematics and its applications, a Sturm–Liouville problem is a second-order linear ordinary differential equation of the form: for given functions , and , together with some boundary conditions at extreme values of . The goals of a given Sturm–Liouville problem are: To find the λ for which there exists a non ...
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] In this case, the change of variable y = ux leads to an equation of the form. which is easy to solve by integration ...
The linear map h → J(x) ⋅ h is known as the derivative or the differential of f at x . When m = n, the Jacobian matrix is square, so its determinant is a well-defined function of x, known as the Jacobian determinant of f. It carries important information about the local behavior of f.
e. In mathematics, a differential-algebraic system of equations ( DAE) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to such a system. The set of the solutions of such a system is a differential algebraic variety, and corresponds to an ideal in a differential algebra of ...
Initial value problem. In multivariable calculus, an initial value problem [a] ( IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem.
Integro-differential equations model many situations from science and engineering, such as in circuit analysis. By Kirchhoff's second law, the net voltage drop across a closed loop equals the voltage impressed . (It is essentially an application of energy conservation .) An RLC circuit therefore obeys.
In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations.. For first-order inhomogeneous linear differential equations it is usually possible to find solutions via integrating factors or undetermined coefficients with considerably less effort, although those methods leverage heuristics that ...
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