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:<math>y = A \cos{x} + B \sin{x} \,</math>,
where ''A'', ''B'' are constants determined from [[boundary
In general, an ''n''-th order equation allows both <math>x</math> and <math>y</math> to be fixed, as well as all the <math>n-1</math> lower order derivatives of <math>y</math>; the remaining equation can be solved (at least conceptionally) for <math>y^{(n)}</math>. If the equation has finite degree <math>d</math>, then we now have a polynomial equation in <math>y^{(n)}</math> with at most <math>d</math> roots. Therefore there can be as many as <math>d</math> possible values for <math>y^{(n)}</math> at any given point and for any possible values of the lower order derivatives, though there may be ranges of these points and values where there are fewer solutions (or none at all). A [[Lipschitz continuity|Lipschitz condition]] must also be satisfied for a solution to exist.
Thus, in the previous example, a second-order, first-degree equation, any point on the plane and any slope through that point can be selected and yield a unique solution (since the single root of <math>y''</math> exists for any value of <math>y</math>). Note in particular that there are an infinity of solutions through any given point; this is a general characteristic of equations of order higher than one
Consider now
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