This preview shows pages 1–7. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Differential Equations & Linear Algebra Lecture 4 Jianli XIE Email: [email protected] Department of Mathematics, SJTU Fall 2010 Contents Exact equations Contents Exact equations Numerical approximations: Euler’s method 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 . 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. 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 ) ....
View
Full
Document
This note was uploaded on 05/17/2011 for the course ME 255 taught by Professor Jianlixie during the Fall '10 term at Shanghai Jiao Tong University.
 Fall '10
 JianliXie

Click to edit the document details