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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
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 ...
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.
The HAM is an analytic approximation method designed for the computer era with the goal of "computing with functions instead of numbers." In conjunction with a computer algebra system such as Mathematica or Maple , one can gain analytic approximations of a highly nonlinear problem to arbitrarily high order by means of the HAM in only a few seconds.
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 (+) + (() +).
Following a win over Arizona, Colorado coach Deion Sanders couldn't help responding to a joke former president Barack Obama made about the Buffaloes.
If you’re stuck on today’s Wordle answer, we’re here to help—but beware of spoilers for Wordle 1242 ahead. Let's start with a few hints.
To the surprise of no one, Bill Belichick reportedly wants to get back to coaching in 2025. (Photo by Andy Lewis/Icon Sportswire via Getty Images) (Icon Sportswire via Getty Images)