80_pdfsam_math 54 differential equation solutions odd

80_pdfsam_math 54 differential equation solutions odd -...

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Unformatted text preview: Chapter 2 11. In this differential equation, M (x, y ) = y 2 + 2xy , N (x, y ) = −x2 . Therefore, ∂M = 2y + 2x, ∂y ∂N = −2x, ∂x and so (∂N/∂x − ∂M/∂y )/M = (−4x − 2y )/(y 2 + 2xy ) = −2/y is a function of y . Then µ(y ) = exp − 2 y dy = exp (−2 ln |y |) = y −2 . Multiplying the differential equation by µ(y ) and solving the obtained exact equation, we get y −2 y 2 + 2xy dx − y −2x2 dy = 0 ⇒ ⇒ ⇒ F (x, y ) = −y −2 x2 dy = y −1 x2 + h(x) ∂ ∂F = y −1 x2 + h(x) = 2y −1x + h (x) = y −2 y 2 + 2xy = 1 + 2xy −1 ∂x ∂x h (x) = 1 ⇒ h(x) = x ⇒ F (x, y ) = y −1x2 + x. Since we multiplied given equation by µ(y ) = y −2 (in fact, divided by y 2 ) to get an exact equation, we could lose the solution y ≡ 0, and this, indeed, happened: y ≡ 0 is, clearly, a solution to the original equation. Thus a general solution is y −1x2 + x = c and y ≡ 0. 13. We will multiply the equation by the factor xn y m and try to make it exact. Thus, we have 2xn y m+2 − 6xn+1 y m+1 dx + 3xn+1 y m+1 − 4xn+2 y m dy = 0. We want My (x, y ) = Nx (x, y ). Since My (x, y ) = 2(m + 2)xn y m+1 − 6(m + 1)xn+1 y m , Nx (x, y ) = 3(n + 1)xn y m+1 − 4(n + 2)xn+1 y m , we need 2(m + 2) = 3(n + 1) 76 and 6(m + 1) = 4(n + 2). ...
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This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at Berkeley.

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