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Higher Order ODEs

# Higher Order ODEs - Chapter 08.05 On Solving Higher Order...

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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 ( 29 ( 29 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 ( 29 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) ( 29 + - - - = = - - - 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 = ( 29 ( 29 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 08.05.1

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08.05.2 Chapter 08.05 ( 29 x z z f z dx dz , , , 2 1 1 2 1 = = (4.1) ( 29 x z z f z dx dz , , , 2 1 2 3 2 = = (4.2) ( 29 ( 29 1 1 0 2 1 1 x f z a z a z a a dx dz n n n n + - - - = - (4.n) Each of the n first order ordinary differential equations are accompanied by one initial condition. These first order ordinary differential equations are simultaneous in nature but can be solved by the methods used for solving first order ordinary differential equations that we have already learned. Example 1 Rewrite the following differential equation as a set of first order differential equations. ( 29 ( 29 7 0 , 5 0 , 5 2 3 2 2 = = = + + - y y e y dx dy dx y d x Solution The ordinary differential equation would be rewritten as follows. Assume , z dx dy = Then dx dz dx y d = 2 2 Substituting this in the given second order ordinary differential equation gives x e y z dx dz - = + + 5 2 3 ( 29 y z e dx dz x 5 2 3 1 - - = - The set of two simultaneous first order ordinary differential equations complete with the initial conditions then is (
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Higher Order ODEs - Chapter 08.05 On Solving Higher Order...

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