hw6-solutions

# X a11 so a21 x a22 y a11 x a12 y y x this

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Unformatted text preview: isﬁes y ′ (t) a21 x + a22 y dy =′ = . dx x (t) a11 x + a12 y Rearranging terms gives (a21 x + a22 y ) + (−a11 x − a12 y ) dy = 0. dx We claim that this is an exact diﬀerential equation. In fact ∂ (a21 x + a22 y ) = a22 ∂y and By assumption, a22 ∂ (−a11 x − a12 y ) = −a11 . ∂x = −a11 , so ∂ ∂ (a21 x + a22 y ) = (−a11 x − a12 y ). ∂y ∂x This shows that the diﬀerential equation is exact. (c) The function ψ (x, y ) = 1 (a21 x2 + 2a22 xy − a12 y 2 ) satisﬁes 2 ∂ ψ (x, y ) = a21 x + a22 y ∂x and ∂ ψ (x, y ) = a22 x − a12 y = −a11 x − a12 y. ∂y Therefore, we have ψ (x, y ) = c along any solution of the diﬀerential equation. In other words, for any solution (x(t), y (t)), we have 1 (a21 x(t)2 + 2a22 x(t)y (t) − a12...
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## This note was uploaded on 02/09/2014 for the course MATH 53 taught by Professor Staff during the Winter '08 term at Stanford.

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