76_pdfsam_math 54 differential equation solutions odd

76_pdfsam_math 54 differential equation solutions odd - M (...

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Chapter 2 (d) We divide the equation, y 2 = kx ,by x and get y 2 /x = k .Thu s , F ( x, y )= y 2 /x and ∂F ∂x = y 2 x 2 , ∂F ∂y = 2 y x 2 y x dx ± y 2 x 2 ² dy =0 2 y x dx = ± y 2 x 2 ² dy 2 xdx = ydy x 2 = y 2 2 + c 1 2 x 2 + y 2 = c. 35. Applying Leibniz’s theorem, we switch the order of diFerentiation (with respect to y )and integration. This yields g 0 = N ( x, y ) x Z x 0 ± ∂y M ( t, y ) ² dt. Therefore, g 0 is diFerentiable (even continuously) with respect to x as a diFerence of two (continuously) diFerentiable functions, N ( x, y ) and an integral with variable upper bound of a continuous function M 0 y ( t, y ). Taking partial derivatives of both sides with respect to x and using fundamental theorem of calculus, we obtain ( g 0 ) ∂x = ∂x N ( x, y ) x Z x 0 ± ∂y M ( t, y ) ² dt = ∂x N ( x, y ) ∂x x Z x 0 ±
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Unformatted text preview: M ( t, y ) dt = x N ( x, y ) y M ( x, y ) = 0 due to (5). Thus ( g ) /x 0 which implies that g does not depend on x (a consequence of mean value theorem). EXERCISES 2.5: Special Integrating Factors, page 71 1. Here M ( x, y ) = 2 y 3 + 2 y 2 and N ( x, y ) = 3 y 2 x + 2 xy . Computing M y = 6 y 2 + 4 y and N x = 3 y 2 + 2 y , we conclude that this equation is not exact. Note that these functions, as well as M itself, depend on y only. Then, clearly, so does the expression ( N/x M/y ) /M , and the 72...
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