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Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011. In 2020, the company was acquired by American educational technology website Course Hero. [3] [4]
In multivariable calculus, an initial value problem [a] (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or other sciences frequently amounts to
In physics, sometimes units of measurement in which c = 1 are used to simplify equations. Time in a "moving" reference frame is shown to run more slowly than in a "stationary" one by the following relation (which can be derived by the Lorentz transformation by putting ∆x′ = 0, ∆τ = ∆t′):
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.
In physics and engineering, the time constant, usually denoted by the Greek letter τ (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system. [1] [note 1] The time constant is the main characteristic unit of a first-order LTI system. It gives speed of the response.
The consequence of this difference is that at every step, a system of algebraic equations has to be solved. This increases the computational cost considerably. If a method with s stages is used to solve a differential equation with m components, then the system of algebraic equations has ms components.
Let us now apply Euler's method again with a different step size to generate a second approximation to y(t n+1). We get a second solution, which we label with a (). Take the new step size to be one half of the original step size, and apply two steps of Euler's method. This second solution is presumably more accurate.
Thus, solutions of the boundary value problem correspond to solutions of the following system of N equations: (;,) = (;,) = (;,) =. The central N−2 equations are the matching conditions, and the first and last equations are the conditions y(t a) = y a and y(t b) = y b from the boundary value problem. The multiple shooting method solves the ...
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