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Unformatted text preview: Chapter 2 = ∂ ∂y x Z x M ( t,y ) dt + g ( y ) = ∂ ∂y x Z x M ( t,y ) dt + N ( x,y ) ∂ ∂y x Z x M ( t,y ) dt = N ( x,y ) . EXERCISES 2.5: Special Integrating Factors 2. This equation is neither separable, nor linear. Since ∂M ∂y = x 1 6 = ∂N ∂x = y, it is not exact either. But M y N x N = x 1 y xy 1 = 1 xy x ( xy 1) = 1 x is a function of just x . So, there exists an integrating factor μ ( x ), which makes the equation exact. 4. This equation is also not separable and not linear. Computing ∂M ∂y = 1 = ∂N ∂x , we see that it is exact. 6. It is not separable, but linear with x as independent variable. Since ∂M ∂y = 4 6 = ∂N ∂x = 1 , this equation is not exact, but it has an integrating factor μ ( x ), because M y N x N = 3 x depends on x only. 8. We find that ∂M ∂y = 2 x, ∂N ∂x = 6 x ⇒ N x M y M = 8 x 2 xy = 4 y depends just on y . So, an integrating factor is μ ( y ) = exp Z 4 y dy = exp ( 4 ln y ) = y 4 . 56 Exercises 2.5 So, multiplying the given equation by y 4 , we get an exact equation 2 xy 3 dx + ( y 2 3 x 2 y 4 ) dy = 0 . Thus, F ( x,y ) = Z 2 xy 3 dx = x 2 y 3 + g ( y ) , ∂F ∂y = 3 x 2 y 4 + g ( y ) = y 2 3 x 2 y 4 ⇒ g ( y ) = y 2 ⇒ g ( y ) = y 1 ....
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This note was uploaded on 02/12/2011 for the course MA 221 taught by Professor Mazmani during the Spring '08 term at Stevens.
 Spring '08
 MAZMANI
 Equations, Factors

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