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 6 4

3. 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

4. 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 =

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 3 2

6. 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

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. 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

9. For example siny = e x + C cosy dy = e x dx Rearrange to make the subject Divide by dx

10. Slope Field Graph

11. 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

12. 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 )

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

14. You try this one when x = 3 y = 0 e y dy = x 2 dx when x = 3 y = 0

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