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In general, the number of arbitrary constants of an ordinary differential equation (ODE) is given by the bespeak of the highest possible derivative.

e.g. $$ycolorred"=f(x)$$ has $colorred extone$ constant of integration, $$ycolorred""+y"=f(x)$$ has $colorred exttwo$, $$ycolorred"""+y""=f(x)$$ has $colorred extthree,$ etc..

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To generalise, intend we"ve an ODE the the form $$a_ny^(colorgreenn)+a_n-1y^^(n-1)+cdots+a_2y""+a_1y"+a_0y=f(x), $$ wherein $a_i$ is a role of $x$.

Then the number of arbitrary constants in the basic solution to this equation is $colorgreenn.$

Notation: $y^(n)=fracd^nydx^n$

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answer Aug 15 "14 at 18:34

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