1. Higher order ODE’s and systems of ODE’s Recall: any higher ODE a system of first order ODEs How to solve? - same as before only more steps

2. Example: Using Leads to

3. Use 4th order Runge-Kutta initial conditions Can be rewritten

4. First calculate k’s Normally, k 1 =f(x,y) Use k 1,1 and k 1,2

5. Normally With two y’s

6. The same is true for k 3 and k 4

7. Now advance both y’s Can extend to many more y’s

8. Back to our example k 1 ’s Because of starting values, all k’s for y 1 are 1 and all k’s for y 2 are 0

11. Another example problem: Deflection of cantilever beam z L y

12. Vertical deflection due to weight J moment of inertia of beam cross section about principle axis E Young’s modulus r density of beam g=-9.8 m/s 2

13. As before, set up as two first order equations Let then

14. Try to solve this three different ways Euler method RK4 Fourth order Adams Need some parameters. Let

15. Euler - so Example calculations

16. Another example: viscous damping If then the analytical solution is

17. Set up system of equations with initial conditions

18. Run the same three methods as before, and compare with analytic solution, given m=1 k=4 F=1