Ordinary differential equation
Summary of exact solutions

Some differential equations have solutions that can be written in an exact and closed form. Several important classes are given here.

In the table below, P (x), Q (x), P (y), Q (y), and M (x, y), N (x, y) are any integrable functions of x, y; b and c are real given constants; C, C,... are arbitrary constants (complex in general). The differential equations are in their equivalent and alternative forms that lead to the solution through integration.

In the integral solutions, λ and ε are dummy variables of integration (the continuum analogues of indices in summation), and the notation ∫ F (λ) dλ just means to integrate F (λ) with respect to λ, then after the integration substitute λ = x, without adding constants (explicitly stated).
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