Explicit differential equation

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Definition

An explicit differential equation is a differential equation where the highest-order derivative is explicitly written as a function of the independent variable, dependent variable, and lower order derivatives. Explicitly, if the independent variable is x, the dependent variable is y, and the order is k, it has the form:

y^{(k)} = G(x,y,y',\dots,y^{(k-1)})

where G is a function of k +1 variables.

Relation with first-degree differential equations

Any explicit differential equation is a first-degree differential equation. Conversely, any first-degree differential equation can be converted to an explicit differential equation via division by a function, though we may need to add back certain solutions that are identically zeros for the function.