By Peter V. O'Neil

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**Extra resources for Advanced Engineering Mathematics, 6th Edition **

**Example text**

1 37 Separable Equations and Integrating Factors We will point out a connection between separable equations and integrating factors. The separable equation y = A x B y is in general not exact. To see this, write it as A x B y −y = 0 so in the present context we have M x y = A x B y and N x y = −1. Now −1 = 0 x and A x B y =A x B y y and in general A x B y = 0. However, y = 1/B y is an integrating factor for the separable equation. If we multiply the differential equation by 1/B y , we get Ax − 1 y =0 By an exact equation because x − 1 By = y A x =0 The act of separating the variables is the same as multiplying by the integrating factor 1/B y .

However, N/ x = M/ y cannot hold for all x y in an entire rectangle in the plane. Hence this differential equation is not exact on any rectangle. 19 Consider ex sin y − 2x + ex cos y + 1 y = 0 With M x y = ex sin y − 2x and N x y = ex cos y + 1, we have M N = ex cos y = x y for all x y . Therefore this differential equation is exact. To find a potential function, set x = ex sin y − 2x and y = ex cos y + 1 32 CHAPTER 1 First-Order Differential Equations Choose one of these equations and integrate it.

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. 13) becomes simple enough to solve if we try as a function of just x or just y. 21 The differential equation x − xy − y = 0 is not exact. 13) is x − = x − xy y Write this as − x = x − xy y −x Now observe that this equation is simplified if we try to find in this event / y = 0 and we are left with just x as just a function of x, because =x This is separable.