Question

Sol. [Key Concept: This is a linear ODE $\frac{dy}{dx} + \frac{y}{x} = x + \sqrt{x^{2} - 1}$ with integrating factor $x$; integrate, apply $y(1) = 1$, then take the greatest integer below $y(\sqrt{5)}$.]

$${\frac{dy}{dx} + \frac{y}{x} = \frac{1}{x - \sqrt{x^{2} - 1}} }{\frac{dy}{dx} + \frac{y}{x} = x + \sqrt{x^{2} - 1} }$$I. $f = e^{lnx} = x$

$${y \cdot x = \int\left( x + \sqrt{x^{2} - 1} \right)xdx }{= \frac{x^{2}}{2} + \frac{\left( x^{2} - 1 \right)^{3/2}}{3} + C }$$Given $y(1) = 1$

$${1 = \frac{1}{2} + C \Rightarrow C = \frac{1}{2} }{\lbrack y(\sqrt{5})\rbrack = \left\lbrack \frac{\sqrt{5}}{2} + \frac{8}{3} + \frac{1}{2\sqrt{5}} \right\rbrack = 3}$$