66_pdfsam_math 54 differential equation solutions odd

66_pdfsam_math 54 differential equation solutions odd -...

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Chapter 2 Integrating M ( x, y ) with respect to x yields F ( x, y )= Z M ( x, y ) dx = Z (cos x cos y +2 x ) dx =c o s y Z cos xdx + Z 2 xdx =s in x cos y + x 2 + g ( y ) . To fnd g ( y ), we compute the partial derivative oF F ( x, y ) with respect to y and compare the result with N ( x, y ). ∂F ∂y = ∂y ± sin x cos y + x 2 + g ( y ) ² = sin x sin y + g 0 ( y )= (sin x sin y +2 y ) g 0 ( y )= 2 y g ( y )= Z ( 2 y ) dy = y 2 . (We take the integration constant C = 0.) ThereFore, F ( x, y )=s in x cos y + x 2 y 2 = c is a general solution to the given equation. 13. In this equation, the variables are y and t , M ( y,t )= t/y , N ( y,t )=1+ln y .S ince ∂M ∂t = ∂t ³ t y ´ = 1 y and ∂N ∂y = ∂y (1 + ln y )=
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Unformatted text preview: Integrating M ( y, t ) with respect to y , we get F ( y, t ) = Z t y dy = t ln | y | + g ( t ) = t ln y + g ( t ) . (rom N ( y, t ) = 1 + ln y we conclude that y > 0.) ThereFore, F t = t [ t ln y + g ( t )] = ln y + g ( t ) = 1 + ln y g ( t ) = 1 g ( t ) = t F ( y, t ) = t ln y + t, and a general solution is given by t ln y + t = c (or, explicitly, t = c/ (ln y + 1)). 62...
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