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Unformatted text preview: Section 2.1: Exact Differential Equations and Integrating Factors Differential Equations can be expressed in derivative form dy dx = f ( x,y ) or the differential form M ( x,y ) dx + N ( x,y ) dy = 0 Definition 1 The total differential dF of the function F is defined by the formula dF ( x,y ) = ∂F ( x,y ) ∂x dx + ∂F ( x,y ) ∂y dy for all (x,y) ∈ D Example 1 Let F be the function F ( x,y ) = xy 2 + 2 x 3 y Then, ∂F ( x,y ) ∂x = y 2 + 6 x 2 y ∂F ( x,y ) ∂y = 2 xy + 2 x 3 and the total differential dF is dF ( x,y ) = ( y 2 + 6 x 2 y ) dx + (2 xy + 2 x 3 ) dy 1 Definition 2 The expression M ( x,y ) dx + N ( x,y ) dy is an exact differential if there exists a function F such that ∂F ( x,y ) ∂x = M ( x,y ) ,and ∂F ( x,y ) ∂y = N ( x,y ) Example 2 The differential equation ( y 2 ) dx + (2 xy ) dy = 0 So, we have ∂F ( x,y ) ∂x = y 2 ∂F ( x,y ) ∂y = 2 xy The function, F ( x,y ) = xy 2 satisfies the partial derivatives and the differential equation given is an exact differential equation.an exact differential equation....
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- Spring '11
- Derivative, Expression, dy, 0 m, equatin