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Published byKenneth Cain Modified over 9 years ago
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Differential Equations There are many situations in science and business in which variables increase or decrease at a certain rate. A differential equation expresses the rate at which one quantity varies in relation to another. If the differential equation is solved then a direct relationship between the two variables can be found.
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Graphing Differential Equations The expression tells us that the graph of 2x if x = 1 then the gradient is if x = 2 then the gradient is if x = 3 then the gradient is How can this be shown on a diagram? y has a gradient of 2 6 4
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Each of the red lines has a gradient equal to twice the x coordinate If the points are joined a family of curves results Slope Field Graph At every point on the curve the gradient is equal to twice the x coordinate ie the gradient equals twice the x coordinate
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If the gradient is x then to find y we : integrate as integration and differentiation are the : reverse of each other = x 2 + C From the slope field graph it can be seen that all the graphs are vertical translations of each other. So C represents a vertical translation y =
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Graphing Differential Equations The expression tells us that the graph of y if y = 1 then the gradient is if y = 2 then the gradient is if y = 3 then the gradient is How can this be shown on a diagram? y has a gradient of 1 3 2
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At every point on the curve the gradient is equal to the y coordinate Each of the red lines has a gradient equal to the y coordinate Slope Field Graph If the points are joined a family of curves results ie the gradient equals the y coordinate
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If the gradient is x then to find y we : integrate as integration and differentiation are the : reverse of each other So how do we integrate an expression where the gradient depends on y rather than x
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Reminder siny = e x + C This must be differentiated using implicit differentiation When differentiating y’s write dy When differentiating x’s write dx Divide by dx Implicit Differentiation
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For example siny = e x + C cosy dy = e x dx Rearrange to make the subject Divide by dx
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Slope Field Graph
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Differentiating siny = e x + C cosy dy = e x dx Reversing the Process cosy dy = e x dx siny = e x + C multiply by cosy multiply by dx integrate Integrating
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Finding the constant C siny = e x + C To find the constant C a boundary condition is needed. If we are told that when x = 0 then y = / 2 then we can find C. siny = e x + C Substitute x = 0 and y = / 2 sin / 2 = e 0 + C So C = 0siny = e x y = sin -1 (e x )
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y= sin -1 (e x ) when x = 0 then y = / 2 Slope Field Graph
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You try this one when x = 3 y = 0 e y dy = x 2 dx when x = 3 y = 0
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Slope Field Graph when x = 3 y = 0
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