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Unformatted text preview: Page 1 of 7 Sample First-Order Differential Equations and their Direction Fields A first-order differential equation has the general form: dy dx = f ( x , y ) where the dependent variable y ( x ) is a function of the independent variable x . Alternatively, if the independent variable is time, it is usually denoted by the variable t , and the unknown x ( t ) is the solution to the differential equation: dx dt = f ( Independent Variable t , Dependent Variable x ) Every first-order differential equation can be represented graphically by a direction field , which at each point ( t , x ) consists of a unit vector whose slope is the derivative dx dt . The train of 5 cars shown below illustrates the first two differential equations. Each car is 25 meters long. 0 25 50 75 100 Direction Field #1: Constants Some things do not change in time. They are constant. Such constant functions are described by the differential equation: dx dt = whose solutions are x ( t ) = c . Direction Field #2: Constant rate of change Some things change at a constant rate. Such functions are described by the differential equation: dx dt = k . The family of solutions consists of the lines x ( t ) = c + k t . Direction Field #3: Semicircles This direction field describes a family of semicircles centered on the origin. The differential equation dy dx = x y is undefined along the horizontal axis, since the derivative is not defined there we cannot divide by zero. The horizontal axis is a forbidden zone . See yellow dotted line. Critical curve is shown as two dotted blue rays. Direction Field #4: Newtons Law of Cooling The rate at which the temperature changes is proportional and opposite to the temperature difference. Hot objects will cool down, and cold objects will warm up. dT dt = k Cooling constant T ( t ) T Temperature Difference Critical Curve: A critical point is a point where the derivative is zero. The set of such points is called a critical curve (or zone) depending on the dimensionality. The horizontal line corresponding to the constant room temperature is thus the critical line for Newtons law of cooling. Critical curves are not usually solutions curves, but this one is. Page 2 of 7 Direct ion Field #5: Compound Growth An initial investment grows at a constant rate proportional to its value. Compound growth is described by the first- order differential equation: dx dt = k Rate of compoud interest x The solution curves are: x c ( t ) = c e kt = x e kt Here the constant c of integration can be interpreted as the initial investment at time 0. In the direction plot above, the doubling time is 10 years. The doubling time is given by solving 2 = e k to obtain: = ln(2) k Direction Field #6: Population Model with Carrying Capacity M A certain species in a closed system satisfies the population model:...
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- Spring '08