mws_gen_ode_spe_higherorder - Chapter 08.05 On Solving...

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08.05.1 Chapter 08.05 On Solving Higher Order Equations for Ordinary Differential Equations After reading this chapter, you should be able to: 1. solve higher order and coupled differential equations , We have learned Euler’s and Runge-Kutta methods to solve first order ordinary differential equations of the form ( ) ( ) 0 0 , , y y y x f dx dy = = (1) What do we do to solve simultaneous (coupled) differential equations, or differential equations that are higher than first order? For example an th n order differential equation of the form ( ) x f y a dx dy a dx y d a dx y d a o n n n n n n = + + + + 1 1 1 1 (2) with 1 n initial conditions can be solved by assuming 1 z y = (3.1) 2 1 z dx dz dx dy = = (3.2) 3 2 2 2 z dx dz dx y d = = (3.3) n n n n z dx dz dx y d = = 1 1 1 (3.n) ( ) + = = x f y a dx dy a dx y d a a dx dz dx y d n n n n n n n 0 1 1 1 1 1 = ( ) ( ) x f z a z a z a a n n n + 1 0 2 1 1 1 (3.n+1) The above Equations from (3.1) to (3.n+1) represent n first order differential equations as follows
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