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In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. [ 1 ]
In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value.
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 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.
Order Equation Application Reference Abel's differential equation of the first kind: 1 = + + + Class of differential equation which may be solved implicitly [1] Abel's differential equation of the second kind: 1
A first-order homogeneous matrix ordinary differential equation in two functions x(t) and y(t), when taken out of matrix form, has the following form: = +, = + where , , , and may be any arbitrary scalars.
In mathematics, a Riccati equation in the narrowest sense is any first-order ordinary differential equation that is quadratic in the unknown function. In other words, it is an equation of the form ′ = + () + () where () and ().
For an arbitrary system of ODEs, a set of solutions (), …, are said to be linearly-independent if: + … + = is satisfied only for = … = =.A second-order differential equation ¨ = (,, ˙) may be converted into a system of first order linear differential equations by defining = ˙, which gives us the first-order system: