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In numerical mathematics, one-step methods and multi-step methods are a large group of calculation methods for solving initial value problems. This problem, in which an ordinary differential equation is given together with an initial condition, plays a central role in all natural and engineering sciences and is also becoming increasingly ...
It costs more time to solve this equation than explicit methods; this cost must be taken into consideration when one selects the method to use. The advantage of implicit methods such as ( 6 ) is that they are usually more stable for solving a stiff equation , meaning that a larger step size h can be used.
For such problems, to achieve given accuracy, it takes much less computational time to use an implicit method with larger time steps, even taking into account that one needs to solve an equation of the form (1) at each time step. That said, whether one should use an explicit or implicit method depends upon the problem to be solved.
The next step is to multiply the above value by the step size , which we take equal to one here: h ⋅ f ( y 0 ) = 1 ⋅ 1 = 1. {\displaystyle h\cdot f(y_{0})=1\cdot 1=1.} Since the step size is the change in t {\displaystyle t} , when we multiply the step size and the slope of the tangent, we get a change in y {\displaystyle y} value.
Single-step methods (such as Euler's method) refer to only one previous point and its derivative to determine the current value. Methods such as Runge–Kutta take some intermediate steps (for example, a half-step) to obtain a higher order method, but then discard all previous information before taking a second step. Multistep methods attempt ...
However, if one searches for real solutions, there are two solutions, √ 2 and – √ 2; in other words, the solution set is {√ 2, − √ 2}. When an equation contains several unknowns, and when one has several equations with more unknowns than equations, the solution set is often infinite. In this case, the solutions cannot be listed.