Exact Differential Equations:

Mdx + Ndy = 0, here M and N are functions of  x and y. If  , then equation  is exact  and  its  solution can be  given  by  ∫ Mdx + ∫ N dy  = c

To find solution of an exact differential equation Mdx + N dy = 0, integrate ∫M dx as if y were constant. Also integrate terms of N which do not contain x w.r.t y. Equate the sum of these integrals to constant.

Example: (x² -ay)dx + (y² -ax)dy = 0.

Solution:         Here M = x² -ay         

                        N = y² -ax

                        ∴ equation is exact

            solution is 

                        x³/3 - ayx + y³/3 = c

                        or x³ -3axy + y³ = 3c.

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