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An ordinary differential equation ( ODE) is an equation containing an unknown function of one real or complex variable x, its derivatives, and some given functions of x. The unknown function is generally represented by a variable (often denoted y ), which, therefore, depends on x. Thus x is often called the independent variable of the equation.
The same illustration for The midpoint method converges faster than the Euler method, as . 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 ...
Matrix differential equation. A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. A matrix differential equation contains more than one function stacked into vector form with a matrix relating the functions to ...
The conclusion of this computation is that =.The exact solution of the differential equation is () =, so () =.Although the approximation of the Euler method was not very precise in this specific case, particularly due to a large value step size , its behaviour is qualitatively correct as the figure shows.
where is the time step (sometimes referred to as ) and is an integer. Multistep methods use information from the previous steps to calculate the next value. In particular, a linear multistep method uses a linear combination of and to calculate the value of for the desired current step. Thus, a linear multistep method is a method of the form.
Cauchy–Euler equation. In mathematics, an Euler–Cauchy equation, or Cauchy–Euler equation, or simply Euler's equation is a linear homogeneous ordinary differential equation with variable coefficients. It is sometimes referred to as an equidimensional equation. Because of its particularly simple equidimensional structure, the differential ...
In mathematics, an ordinary differential equation ( ODE) is a differential equation (DE) dependent on only a single independent variable. As with other DE, its unknown (s) consists of one (or more) function (s) and involves the derivatives of those functions. [1] The term "ordinary" is used in contrast with partial differential equations which ...
Heun's method. In mathematics and computational science, Heun's method may refer to the improved [1] or modified Euler's method (that is, the explicit trapezoidal rule [2] ), or a similar two-stage Runge–Kutta method. It is named after Karl Heun and is a numerical procedure for solving ordinary differential equations (ODEs) with a given ...
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