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lecture_10 (dragged) 2

lecture_10 (dragged) 2 - x y μ x y N x y = x y 2 x y y 2...

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MA 36600 LECTURE NOTES: WEDNESDAY, FEBRUARY 4 3 Integrating Factors. Say that we have a first order di ff erential equation dy dx = G ( x, y ) where G ( x, y ) = M ( x, y ) N ( x, y ) . If μ = μ ( x, y ) is any nonzero function, we have G ( x, y ) = μ ( x, y ) · M ( x, y ) μ ( x, y ) · N ( x, y ) = μ ( x, y ) M ( x, y ) dx + μ ( x, y ) N ( x, y ) dy = 0 . This is an exact equation when y μ M = x μ N . Such a function μ = μ ( x, y ) is called an integrating factor . If the equation M dx + N dy = 0 is an exact equation, we may choose μ ( x, y ) = 1. In general, we seek an integrating factor which will make the equation exact. Example. Consider the di ff erential equation dy dx = 3 x y + y 2 x 2 + x y = 3 x y + y 2 dx + x 2 + x y dy = 0 . We saw above that this equation is not an exact equation. Consider the function μ ( x, y ) = x . Then we have μ ( x, y ) · M ( x, y ) = 3 x 2 y + x y 2 μ ( x, y ) · N ( x, y ) = x 3 + x 2 y = y μ M = 3 x 2 + 2 x y x μ N = 3 x 2 + 2 x y Hence we have the exact di ff erential equation 3 x 2 y + x y 2 dx + x 3 + x 2 y dy = 0 . Integrating factors need not be unique. Indeed, consider the function μ ( x, y ) = 1 2 x 2 y + x y 2 . Then we have μ ( x, y ) · M ( x, y ) = 3 x + y 2 x 2 + x y μ ( x, y ) · N ( x, y
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Unformatted text preview: x y μ ( x, y ) · N ( x, y ) = x + y 2 x y + y 2 = ⇒ ∂ ∂y ° μ M ± = − 1 (2 x + y ) 2 ∂ ∂x ° μ N ± = − 1 (2 x + y ) 2 Hence we have the exact diFerential equation 3 x + y 2 x 2 + x y dx + x + y 2 x y + y 2 dy = 0 . Linear Equations Revisited. Consider the frst order linear equation dy dt + p ( t ) y + g ( t ) . We have seen be±ore that this equation has an integrating ±actor. We explain how this is related to the concept o± integrating ±actor defned in the previous lecture. We may multiply by the diFerential dt to fnd the diFerential equation M ( t, y ) dt + N ( t, y ) dy = 0 in terms o± the ±unctions M ( t, y ) = g ( t ) − p ( t ) y and N ( t, y ) = − 1 . This diFerential equation is not an exact equation because ∂M ∂y = − p ( t ) , ∂N ∂t = 0 = ⇒ ∂M ∂y ° = ∂N ∂t ....
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