IntroDiffEquations

IntroDiffEquations - Differential Equations In physics,...

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Differential Equations In physics, engineering, chemistry, economics, and other sciences mathematical models are built that involve rates at which things happen. These models are equations and the rates are derivatives. Equations containing derivatives are called differential equations . Examples : 1) 100 grams of cane sugar in water are being converted into dextrose at a rate that is proportional to the amount unconverted. Find the differential equation expressing the rate of conversion after t minutes. Solution : Let q be the grams of sugar converted in t minutes, then (100 – q) is the number of grams unconverted and the rate of conversion is given by dq dt = k(100 ! q) where k is the constant of proportionality. 2) A curve is defined by the condition that at each of its points (x,y), is slope dy/dx is equal twice the sum of the coordinates of the point, find the differential equation that defines the curve. Solution : dy dx = 2(x + y) The following are more examples of differential equations: 1) dy dx = cosx or y' = cosx 2) d 2 y dx 2 + k 3 y = 0 or ! ! y + k 3 y = 0 3) (x 2 + y 2 )dx = 2xdy 4) ! u ! t = h 2 ( ! 2 u ! x 2 + ! 2 u ! y 2 ) 5) ! 2 w ! x 2 + ! w ! y = 0 6) d 2 s dt 2 ! " # $ % 6 xy ds dt + s = 0 7) d 3 x dy 3 + x dx dy = 4xy
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8) d 5 y dx 5 + x dy dx ! " # $ % 3
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This note was uploaded on 07/23/2011 for the course MAP 2302 taught by Professor Staff during the Fall '08 term at FIU.

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IntroDiffEquations - Differential Equations In physics,...

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