MA232_Project_Fall08 - MA232 Numerical Project Fall 2008...

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MA232 Numerical Project Fall 2008 Due 9/19 1 Reduction of Order The numerical methods that we will look at this week are good for solving 1 st order DE’s. However, we wish to model 2 nd order DE’s because they are more interesting. This is O.K. because we can reduce 2 nd order DE’s into systems of 1 st order DE’s. An example: Hooke’s law is m d 2 x dt 2 = - kx , which is 2 nd order. If we define dx dt = V x , putting this into above, Hooke’s law is now m dV x dt = - kx . These boxed equations represent our new 1 st order system of DE’s. Note that before we only had one dependent variable: x . Now, in our system, we have two dependent variables: x and V . For practice, reduce the following 2 nd order DE’s into systems of 1 st order DE’s. Where applicable, please use the notation dt = ω , dx dt = V x , and dy dt = V y . (a) Harmonic Oscillator (mass-spring-damper system): d 2 x dt 2 + c dx dt + kx =0 (reduce to two 1 st order DE’s) (b) Pendulum: d 2 θ dt 2 + g L sin ( θ )=0 (reduce to two 1 st order DE’s) (c) Projectile Motion: (note: V x
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