lecture_10 (dragged) 2

lecture_10 (dragged) 2 - x y ( x, y ) N ( x, y ) = x + y 2...

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MA 36600 LECTURE NOTES: WEDNESDAY, FEBRUARY 4 3 Integrating Factors. Say that we have a frst order diFerential equation dy dx = G ( x, y )w h e r e G ( x, y )= M ( x, y ) N ( x, y ) . μ = μ ( x, y ) is any nonzero ±unction, 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 ° μM ± = ∂x ° μN ± . Such a ±unction μ = μ ( x, y ) is called an integrating factor . I± the equation Mdx + Ndy = 0 is an exact equation, we may choose μ ( x, y ) = 1. In general, we seek an integrating ±actor which will make the equation exact. Example. Consider the diFerential equation dy dx = 3 xy + y 2 x 2 + xy = ² 3 xy + y 2 ³ dx + ² x 2 + xy ³ dy =0 . We saw above that this equation is not an exact equation. Consider the ±unction μ ( x, y )= x .Thenw ehav e μ ( x, y ) · M ( x, y )=3 x 2 y + xy 2 μ ( x, y ) · N ( x, y )= x 3 + x 2 y = ∂y ° μM ± =3 x 2 +2 xy ∂x ° μN ± =3 x 2 +2 xy Hence we have the exact diFerential equation ² 3 x 2 y + xy 2 ³ dx + ² x 3 + x 2 y ³ dy =0 . Integrating ±actors need not be unique. Indeed, consider the ±unction μ ( x, y )= 1 2 x 2 y + xy 2 . Then we have μ ( x, y ) · M ( x, y )= 3 x + y 2 x 2
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Unformatted text preview: x y ( x, y ) N ( x, y ) = x + y 2 x y + y 2 = y M = 1 (2 x + y ) 2 x 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 beore 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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