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  2. Numerical methods for ordinary differential equations - Wikipedia

    en.wikipedia.org/wiki/Numerical_methods_for...

    where is a function : [,), and the initial condition is a given vector. 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 ...

  3. List of nonlinear ordinary differential equations - Wikipedia

    en.wikipedia.org/wiki/List_of_nonlinear_ordinary...

    Important in complex analysis and geometric function theory [15] Logistic differential equation (sometimes known as the Verhulst model) 2 = (()) Special case of the Bernoulli differential equation; many applications including in population dynamics [16] Lorenz attractor: 1

  4. Heun's method - Wikipedia

    en.wikipedia.org/wiki/Heun's_method

    The accuracy of the Euler method improves only linearly with the step size is decreased, whereas the Heun Method improves accuracy quadratically . [5] The scheme can be compared with the implicit trapezoidal method , but with f ( t i + 1 , y i + 1 ) {\displaystyle f(t_{i+1},y_{i+1})} replaced by f ( t i + 1 , y ~ i + 1 ) {\displaystyle f(t_{i+1 ...

  5. Finite difference method - Wikipedia

    en.wikipedia.org/wiki/Finite_difference_method

    For example, consider the ordinary differential equation ′ = + The Euler method for solving this equation uses the finite difference quotient (+) ′ to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get (+) + (() +).

  6. Ordinary differential equation - Wikipedia

    en.wikipedia.org/wiki/Ordinary_differential_equation

    Some ODEs can be solved explicitly in terms of known functions and integrals. When that is not possible, the equation for computing the Taylor series of the solutions may be useful. For applied problems, numerical methods for ordinary differential equations can supply an approximation of the solution.

  7. Adaptive step size - Wikipedia

    en.wikipedia.org/wiki/Adaptive_step_size

    where y and f may denote vectors (in which case this equation represents a system of coupled ODEs in several variables). We are given the function f(t,y) and the initial conditions (a, y a), and we are interested in finding the solution at t = b. Let y(b) denote the exact solution at b, and let y b denote the solution that we compute.

  8. When are the second College Football Playoff rankings ...

    www.aol.com/second-college-football-playoff...

    The second College Football Playoff Rankings will be released in a little more than 24 hours. If the committee members stick to their approach from the first rankings, then it's not only about how ...

  9. System of differential equations - Wikipedia

    en.wikipedia.org/wiki/System_of_differential...

    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: