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

Example: Using Leads to

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

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

Normally With two y’s

The same is true for k 3 and k 4

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

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

Another example problem: Deflection of cantilever beam z L y

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

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

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

Euler - so Example calculations

Another example: viscous damping If then the analytical solution is

Set up system of equations with initial conditions

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