lecture4 - Dierential Equations Linear Algebra Lecture 4...

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Differential Equations & Linear Algebra Lecture 4 Jianli XIE Email: [email protected] Department of Mathematics, SJTU Fall 2010
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Contents Exact equations
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Contents Exact equations Numerical approximations: Euler’s method
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Exact equations Consider the differential equation M ( x, y ) + N ( x, y ) y = 0 . (1) Suppose that there is a function ψ ( x, y ) such that ∂ψ ∂x ( x, y ) = M ( x, y ) , ∂ψ ∂y ( x, y ) = N ( x, y ) (2) then equation (1) is called an exact equation . To solve the equation, we write in into M ( x, y )+ N ( x, y ) y = ∂ψ ∂x + ∂ψ ∂y dy dx = d dx ψ ( x, y ) = 0 .
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Exact equations Therefore, the solution is ψ ( x, y ) = c, which implicitly defines the function y = y ( x ) . From above derivation, we see that recognizing the required function ψ ( x, y ) is the key point to confirm the equation is exact and to find the solution. There is an equivalent condition to (2) M y ( x, y ) = N x ( x, y ) (3) which is easy to check.
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Exact equations Now suppose that the equation is indeed exact, that is, (3) holds true, we want to find the function ψ ( x, y ) to solve the equation. We start from condition (2) ψ x ( x, y ) = M ( x, y ) , ψ y ( x, y ) = N ( x, y ) . Integrating the first equality with respect to x , holding y constant, we obtain ψ ( x, y ) = x M ( x, y ) dx + h ( y ) , (4) where h ( y ) is a function to be determined by the second equality in (2).
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