# 2 - Last Day Exact Equations What are they Basically...

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Last Day? Exact Equations What are they??? Basically, differential equations of the form: M ( x , y ) dx N ( x , y ) dy 0 where: M y N x . What does M y N x ensure? Math 2860 Summer 2010: Lecture 2 Page 1 of 25

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How to Solve??? (We’re looking for a function f(x, y) such that f x = M and f y = N.) Set f x = M and integrate with respect to x to get an expression for f . Differentiate your expression with respect to y to get an expression for f y . Set f y = N . Solve for g ( y ). f ( x, y ) = C is the general solution. Math 2860 Summer 2010: Lecture 2 Page 2 of 25
Example 1: Solve . (6 xy 1) dx (3 x 2 y ) dy 0 Solution: Is it exact? Set f x = M and integrate with respect to x to get an expression for f . Differentiate your expression with respect to y to get an expression for f y . Set f y = N . Solve for g ( y ). f ( x, y ) = C is the general solution. Math 2860 Summer 2010: Lecture 2 Page 3 of 25

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Example 2: Solve y 2 x y 2 y x . Solution: Is it exact? Set f x = M and integrate with respect to x to get an expression for f .
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2 - Last Day Exact Equations What are they Basically...

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