lecture notes set 13 - ME 218 ENGINEERING COMPUTATIONAL...

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ME 218: ENGINEERING COMPUTATIONAL METHODS Notes Set 13: Fall 2011 Higher order equations: The above techniques hold good for first order ODEs. Higher order ODE’s need to be converted to 1 st order ODEs before any one of the above formulations can be applied. Let us assume we have an ODE, with the n th order derivative present in the equation, as follows: 0 ) , ( ... 1 2 2 2 1 1 1 = + + + + + - - - y x f dx dy a dx y d a dx y d a dx y d a n n n n n n , where a i is the coefficient of the corresponding derivative. This equation can be reduced into a set of 1 st order ODEs by defining a set of new dependent variables as follows: 1 2 3 2 2 1 2 2 1 1 z y z dx dy z dx y d z dx y d z dx y d n n n n n n = = = = = - - - - - M . There are ( n – 1) such “dummy” variables, and the “dummy” equations corresponding to these variables are: 2 1 3 2 1 2 1 z dx dz z dx dz z dx dz z dx dz n n n n = = = = - - - M
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Thus, the “main” equation reduces to 0 ) , ( ... 1 2 1 3 2 1 = + + + + + - z x f z a z a z a dx dz a n n n n , which is a 1 st order ODE! By including the “dummy” equations we have a system of n 1 st order ODEs with the independent variable as x and the n dependent variables as y (which is really z 1 ), z 2 , …, z n- 1 and z n . Notice that the order of the highest derivative determines
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