1. 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

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

3. 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

4. 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 )

5. 1 st Order Differential Eqns Separating the Variables (C4) 1) Collect the x’s on one side and the y’s and the on the other side. 2) Multiply both side by dx and integrate. 3) Simplify the resulting equation. y fns join the dy and x fns join the dx usually by Cross Multiplying N.B y functions are integrated w.r.t y x functions are integrated w.r.t x

6. Ex1 Solve the differential equation Multiply both sides by y Multiply both side by dx and integrate Find C by substituting in 2 boundary conditions

7. Differential equations where the variables cannot be separated Ex1 If we try to rearrange so that the x`s and dx`s are on one side and the y`s and dy`s are on the other it is impossible. Try it!!

8. This is an example of a 1 st order Linear Differential Equation. It can be solved however by multiplying every thing by x 3.

9. What could we differentiate that would give this expression. It must contain a mixture of x`s and y`s…….. Differentiating x 3 y x 3 dy + 3x 2 ydx using implicit differentiation divide by dx

10. So if we integrate we obtain x 3 y Integrate both sides

11. How do we find what to multiply by to make the LHS into an expression which can be integrated by inspection. THE INTEGRATING FACTOR !!!!!! If Then the integrating factor =

12. Differential equations where the variables cannot be separated Steps 1.Rearrange in the form 2.Find the integrating factorI.F. = 3.Multiply every term by the I.F. 4.Integrate each side – the integral of the L.H.S. is I.F × y

13. 1. Rearrange in the form Divide by x P(x) =and Q(x) = 1

14. 3. Multiply every term by the I.F. 4. Integrate each side – the integral of the L.H.S. is I.F × y P(x) = I.F = 2. Find the integrating factorI.F. =

15. How can you spot what the integral of the LHS is??? If we integrate we obtain x 3 y This is the same as I.F. × y So the integral of the LHS is I.F. × y

16. + cotx y = cosecxx = 0 when y = 0 P(x) = cotx Q(x) = cosecx I.F = = sinx + sinx cotx y = sinx cosecx 1. Rearrange in the form 2. Find the integrating factorI.F. = 3. Multiply every term by the I.F.

17. + sinx cotx y = sinx cosecx Simplify + sinx y = sinx+ cosx y = 1 Integrating the LHS gives sinx y sinx y = = x + c i.e I.F × y 4. Integrate each side – the integral of the L.H.S. is I.F × y

18. sinx y = = x + c Substitute x = 0 and y = 0gives c = 0 sinx y = xSo y =