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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 ...
Differential equations are prominent in many scientific areas. Nonlinear ones are of particular interest for their commonality in describing real-world systems and how much more difficult they are to solve compared to linear differential equations.
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.
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.
Although this is defined using a particular coordinate system, the transformation law relating the ξ i and the x i ensures that σ P is a well-defined function on the cotangent bundle. The function σ P is homogeneous of degree k in the ξ variable. The zeros of σ P, away from the zero section of T ∗ X, are the characteristics of P.
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)
Federal Reserve Chair Jerome Powell said he won't step down if President-elect Donald Trump, who has previously criticized Powell's performance, asks him to resign.
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 (+) + (() +).