First-degree differential equation

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Definition

A first-degree differential equation is a differential equation that is linear in its highest-order derivative. Explicitly, of the independent variable is x, the dependent variable is y, and the order is kit has the form:

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

Relation with explicit differential equations

First-degree differential equations are typically solved by converting them to explicit differential equations. This simply involves dividing by P. The explicit differential equation form is:

y^{(k)} = \frac{Q(x,y,y',\dots,y^{(k-1)})}{P(x,y,y',\dots,y^{(k-1)})}

However, when doing this division, we may throw away some solutions where P(x,y,y',\dots,y^{(k-1)}) = Q(x,y,y',\dots,y^{(k-1)}) = 0. These solutions can be checked for separately, but we can note that these are differential equations of a lower order (specifically, order at most k - 1) hence presumably easier to solve.