nagle_differential_equations_ISM_Part13

nagle_differential_equations_ISM_Part13 - Chapter 2 = ∂...

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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.

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nagle_differential_equations_ISM_Part13 - Chapter 2 = ∂...

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