By Peter V. O'Neil

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10 Linear differential equations arise in many contexts. 11, involving estimation of time of death, involved a separable differential equation which is also linear and could have been solved using an integrating factor. 16 (A Mixing Problem) Sometimes we want to know how much of a given substance is present in a container in which various substances are being added, mixed, and removed. Such problems are called mixing problems, and they are frequently encountered in the chemical industry and in manufacturing processes.

The function therefore enabled us to solve a nonexact equation by solving an exact one. This idea is worth pursuing, and we begin by giving a name to . 5 Let M x y and N x y be defined on a region R of the plane. Then x y is an integrating factor for M + Ny = 0 if x y = 0 for all x y in R, and M + Ny = 0 is exact on R. How do we find an integrating factor for M + Ny = 0? 13) 34 CHAPTER 1 First-Order Differential Equations in this region. This is a starting point. Depending on M and N , we may be able to determine from this equation.

If we cannot find an integrating factor that is a function of just x or just y, then we must try something else. 13) and be observant. 22 Consider 2y2 − 9xy + 3xy − 6x2 y = 0. This is not exact. 14) / y = 0 and we obtain 3xy − 6x2 x + 3y − 12x = 4y − 9x which cannot be solved for as just a function of x. Similarly, if we try = y , so / x = 0, we obtain an equation we cannot solve. We must try something else. 14) involves only integer powers of x and y. This suggests that we try x y = xa yb . 14) and attempt to choose a and b.