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Mixing problems are an application of differential equations that can be solved using the integrating factor method. Simply put, the integrating factor is a function that we multiply both sides of the differential equation by to make it easier to solve.
We can solve these differential equations using the technique of an integrating factor. We multiply both sides of the differential equation by the integrating factor I which is defined as I = e P DX. In = IQ DX since d DX (In) = I Dy DX + IPY by the product rule.
An integrating factor is a function by which an ordinary differential equation can be multiplied in order to make it integrable. For example, a linear first-order ordinary differential equation of type. (1)
Suggested clip The Integrating Factor Method — YouTubeYouTubeStart of suggested client of suggested clip The Integrating Factor Method — YouTube
In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. ... This is especially useful in thermodynamics where temperature becomes the integrating factor that makes entropy an exact differential.
INTEGRATING FACTOR is found in May 1845 in a paper by Sir George Gabriel Stokes published in the Cambridge Mathematical Journal [University of Michigan Historical Math Collection].
It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus when multiplying through by an integrating factor allows an inexact differential to be made into an exact differential (which can then be integrated to give a scalar field).
We multiply both sides of the differential equation by the integrating factor I which is defined as I = e P DX. In = IQ DX since d DX (In) = I Dy DX + IPY by the product rule. As both I and Q are functions involving only x in most of the problems you are likely to meet, IQ DX can usually be found.
Because when you multiply the equation by the integrating factor including an arbitrary constant, it just becomes a common constant factor across the whole equation. It can therefore be cancelled without any change to the solution.
Suggested clip Exact Differential Equations — YouTubeYouTubeStart of suggested client of suggested clip Exact Differential Equations — YouTube
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