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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.
Third-order methods can be generically written ... (1996), Solving ordinary differential equations II: Stiff and differential-algebraic problems, Berlin, New York ...
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 (+) + (() +).
In calculus, a branch of mathematics, the third derivative or third-order derivative is the rate at which the second derivative, or the rate of change of the rate of change, is changing. The third derivative of a function y = f ( x ) {\displaystyle y=f(x)} can be denoted by
This is a third-order non-linear ordinary differential equation which can be solved numerically, e.g. with the shooting method. The boundary condition at infinity is converted to f ″ ( 0 ) = 0.332043934904293 {\displaystyle f''(0)=0.332043934904293} .
In mathematics, and in particular in the theory of solitons, the Dym equation (HD) is the third-order partial differential equation u t = u 3 u x x x . {\displaystyle u_{t}=u^{3}u_{xxx}.\,} It is often written in the equivalent form for some function v of one space variable and time
The A-stability concept for the solution of differential equations is related to the linear autonomous equation ′ =. Dahlquist (1963) proposed the investigation of stability of numerical schemes when applied to nonlinear systems that satisfy a monotonicity condition.
When converted to an equivalent system of three ordinary first-order non-linear differential equations, jerk equations are the minimal setting for solutions showing chaotic behaviour. This condition generates mathematical interest in jerk systems. Systems involving fourth-order derivatives or higher are accordingly called hyperjerk systems. [1]