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This system can now be solved using dsolve.
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The system was solved by coupling the ODEs above and splitting the set into three subsets, which are mutually uncoupled.
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This makes it possible to solve each ODE separately.
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When computing the characteristic strip, by default the strip is simplified taking the PDE itself into account, aiming at diminishing occurrences of derivatives of the unknown of the problem. This is useful in that it makes easier the integration of the resulting ODE system. On the other hand, there are situations in which this simplification is undesired, for example: when deriving the PDE behind a given characteristic strip. Consider for instance
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The corresponding characteristic strip without simplifying it taking PDE into account is given by
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Departing from this result it is possible to re-obtain PDE (this is possible only when the characteristic strip was constructed without simplifying using the PDE itself, as it is the case above). For that purpose, pass the characteristic strip as first argument and the keyword reverse
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Note the arbitrary constant that appears in this result: for any value of this constant, the characteristic strip is the same.