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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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- Spring '09