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Differential Equations

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.

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Differential Equations

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  1. 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.

  2. 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 4 6

  3. ie the gradient equals twice the x coordinate Slope Field Graph Each of the red lines has a gradient equal to twice the x coordinate If the points are joined a family of curves results At every point on the curve the gradient is equal to twice the x coordinate

  4. y = If the gradient is x then to find y we : integrate as integration and differentiation are the : reverse of each other = x2 + 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

  5. 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 2 3

  6. ie the gradient equals the y coordinate Slope Field Graph If the points are joined a family of curves results Each of the red lines has a gradient equal to the y coordinate At every point on the curve the gradient is equal to the y coordinate

  7. 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

  8. Implicit Differentiation Reminder siny = ex + C This must be differentiated using implicit differentiation • When differentiating y’s write dy • When differentiating x’s write dx • Divide by dx

  9. Divide by dx Rearrange to make the subject Implicit Differentiation For example siny = ex + C cosy dy = ex dx

  10. Slope Field Graph

  11. multiply by dx multiply by cosy integrate Reversing the Process Integrating Differentiating siny = ex + C cosy dy = ex dx cosy dy = ex dx siny = ex + C

  12. Finding the constant C siny = ex + CTo find the constant C a boundary condition is needed. If we are told that when x = 0 then y = p/2then we can find C. siny = ex + C Substitute x = 0 and y = p/2 sin p/2 = e0 + C So C = 0 siny = ex y = sin-1(ex)

  13. Slope Field Graph y= sin-1(ex) when x = 0 then y = p/2

  14. when x = 3 y = 0 You try this one ey dy = x2 dx when x = 3 y = 0

  15. Slope Field Graph when x = 3 y = 0

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