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(C) The given differential equation is $(x^2 - x\sqrt{x^2-1})dy = (x - y(x - \sqrt{x^2-1}))dx$.Rearranging,we get $\frac{dy}{dx} + \frac{x - \sqrt{x^2

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$3$

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(C) The given differential equation is $(x^2 - x\sqrt{x^2-1})dy = (x - y(x - \sqrt{x^2-1}))dx$.Rearranging,we get $\frac{dy}{dx} + \frac{x - \sqrt{x^2-1}}{x(x - \sqrt{x^2-1})}y = \frac{x}{x(x - \sqrt{x^2-1})}$.This simplifies to $\frac{dy}{dx} + \frac{1}{x}y = \frac{1}{x - \sqrt{x^2-1}}$.Multiplying by $x + \sqrt{x^2-1}$,we get $\frac{dy}{dx} + \frac{1}{x}y = x + \sqrt{x^2-1}$.The integrating factor $IF = e^{\int \frac{1}{x} dx} = x$.The solution is $y \cdot x = \int x(x + \sqrt{x^2-1}) dx = \int (x^2 + x\sqrt{x^2-1}) dx$.$xy = \frac{x^3}{3} + \frac{1}{3}(x^2-1)^{3/2} + C$.Given $y(1) = 1$,we have $1(1) = \frac{1}{3} + 0 + C$,so $C = \frac{2}{3}$.Thus,$y = \frac{x^2}{3} + \frac{(x^2-1)^{3/2}}{3x} + \frac{2}{3x}$.For $x = \sqrt{5}$,$y(\sqrt{5}) = \frac{5}{3} + \frac{(4)^{3/2}}{3\sqrt{5}} + \frac{2}{3\sqrt{5}} = \frac{5}{3} + \frac{8}{3\sqrt{5}} + \frac{2}{3\sqrt{5}} = \frac{5}{3} + \frac{10}{3\sqrt{5}} = \frac{5}{3} + \frac{2\sqrt{5}}{3} = \frac{5 + 2(2.236)}{3} \approx \frac{9.472}{3} \approx 3.157$.The greatest integer less than or equal to $3.157$ is $3$.

Let $y = y(x)$ be the solution of the differential equation $(x^2 - x\sqrt{x^2-1})dy + (y(x - \sqrt{x^2-1}) - x)dx = 0, x \geq 1$. If $y(1) = 1$,then the greatest integer less than or equal to $y(\sqrt{5})$ is . . . . . . .

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