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Jhenrique

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In summary, the conversation discusses the concept of explicitly representing an implicit ODE in terms of its higher-order derivatives, which is done for convenience and to simplify solving the equation. This is a common convention and allows for more manageable and known techniques to be applied in solving the equation.

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Jhenrique

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Mark44

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What they are saying is that starting with an equation F( ... ) = 0 that involves x, y(x), y'(x), and y''(x), where y''(x) is given implicitly, a new equation can be written that gives y''(x) explicitly as a function of x and the lower-order derivatives.Jhenrique said:

A very simple example would be y'' - 2y' + 2y = 0. Here the left side is F(x, y, y', y'').

With y'' given explicitly, we have y'' = 2y' - 2y. Here the right side is f(x, y, y').

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Jhenrique

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Mark44 said:What they are saying is that starting with an equation F( ... ) = 0 that involves x, y(x), y'(x), and y''(x), where y''(x) is given implicitly, a new equation can be written that gives y''(x) explicitly as a function of x and the lower-order derivatives.

A very simple example would be y'' - 2y' + 2y = 0. Here the left side is F(x, y, y', y'').

With y'' given explicitly, we have y'' = 2y' - 2y. Here the right side is f(x, y, y').

?

You gave an example for my question, but not an answer...

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Mark44

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AFAIK, this is partly convention and partly convenience.

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[tex]F(t,y,y^\prime,...,y^{(n)}) = 0[/tex]

is pretty horrible and there are very little techniques known. On the other hand,

[tex]y^{(n)} = f(t,y,...,y^{(n-1)})[/tex]

is way more manageable and more results about these equations are known. Also, most equations in practice show up in this form.

An explicit ODE is typically represented with y of a higher grade because it allows for a more accurate and efficient solution. This representation allows for the use of higher order methods, which can provide a more precise solution compared to lower order methods. Additionally, it can also reduce the number of steps needed to solve the ODE, making it more computationally efficient.

The main purpose of using y of a higher grade in an explicit ODE is to improve the accuracy and efficiency of the solution. Higher order methods can provide a more precise solution and reduce the computational time needed to solve the ODE. This is especially important for complex and large-scale ODEs.

Using y of a higher grade in an explicit ODE can greatly improve the accuracy of the solution. This is because higher order methods can capture finer details and fluctuations in the solution that lower order methods may miss. It can also reduce the error in the solution, resulting in a more reliable and precise solution.

While using y of a higher grade can greatly improve the accuracy and efficiency of the solution, it may also require more computational resources. Higher order methods often involve more complex calculations, which can increase the computational time and memory needed to solve the ODE. Therefore, it is important to consider the trade-off between accuracy and computational cost when choosing the order of y in an explicit ODE.

Using y of a higher grade is necessary in an explicit ODE when the solution requires a high level of accuracy or when the ODE is complex and sensitive to changes in the initial conditions. This is often the case in scientific and engineering applications, such as in climate modeling, chemical kinetics, and structural analysis. In these scenarios, using y of a higher grade can greatly improve the reliability and usefulness of the solution.

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