Exact differential equations of the first order

The reason many differential equations are so difficult to solve is due to the fact that they have been formed by the elimination of constants as well as by the elision of some common factor from the primitive. Such an equation, therefore, does not actually represent the complete or total differential of the original equation or primitive. The equation is then said to be inexact. On the other hand, an exact differential equation is one that has been obtained by the differentiation of a function of x and y and performing no other operation involving x and y. 

Easy tests have been described, on page 77, to determine 

This last result was oalled the criterion of integrability, because, if an equation satisfies the test, the integration can be readily performed by a direct process. This is not meant to imply that only such equations oan be integrated as satisfy the test, for many equations which do not satisfy the test can be solved in other ways. 

Examples.—(1) Apply the test to the equations, ydx + xdy = 0, and ydx - xdy = 0. In the former, M y, N x . . dMfdy = 1, dNfdx 1 . . dMfdy = dN/dx. The test is, therefore, satisfied and the equation is exact. In the other equation, M = y, N - x,. . dMfdy = 1, dNfdx = - 1. This does not satisfy the test. In oonsequence, the equation cannot be solved by the method for exact differential equations. 

We must remember that M is the differential coefficient of u with respect to x, y being constant, and N is the differential coefficient of u with respect to y, x being constant. Hence we may integrate Mdx on the supposition that y is constant and then treat Ndy as if x were a constant. The oomplete solution of the whole equation is obtained by equating the sum of these two integrals to an undetermined constant. The oomplete integral is 

---