Transforming higher-order ODEs

In this example we will consider the transformation of a second-order ODE to a system of two first-order ODEs. The second-order differential equation is given 1  

Subsequently, the second-order equation is transformed to the system 

This system of first-order differential equations is integrated to solve file second-order ODE Equation (6.34), using any of the metiiods described in the previous chapters. 

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The integration of a DAE system can be performed by transformation in an ODE system. It is worthy to note that this operation might be confronted with the index problem. Index is the minimum number order of differentiation needed to transform a DAE system into a set of first-order ODEs. Problems of index one can be solved by means of standard differentiation methods. When the index is higher than one then the DAE system needs a special treatment. Modem codes have capabilities for automatic detection of index higher-than-one, diagnose the problem and suggest modifications. 

It is clear from this example that the Laplace transform solution for complex or repeated roots can be quite cumbersome for transforms of ODEs higher than second order. In this case, using numerical simulation techniques may be more efficient to obtain a solution, as discussed in Chapters 5 and 6. 

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