Question

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 $y(\sqrt{5})$ is ________.

Hint:

Rearrange the differential equation into the linear form y' + P(x)y = Q(x). Rationalize the denominator of the term on the right-hand side to simplify integration.

Answer 1

Correct Answer: 3

The goal is to solve the given differential equation and then find the floor value of the solution at $x = \sqrt{5}$.

Step 1: Simplify the differential equation.
The equation is: $(x^2 - x\sqrt{x^2 - 1}) dy + (y(x - \sqrt{x^2 - 1}) - x) dx = 0$.
Divide by $dx$ and rearrange:
$x(x - \sqrt{x^2 - 1}) \frac{dy}{dx} + y(x - \sqrt{x^2 - 1}) = x$.
Now, divide the entire equation by $(x - \sqrt{x^2 - 1})$:
$x \frac{dy}{dx} + y = \frac{x}{x - \sqrt{x^2 - 1}}$.

Step 2: Rationalize the right-hand side.
$$\frac{x}{x - \sqrt{x^2 - 1}} = \frac{x(x + \sqrt{x^2 - 1})}{(x - \sqrt{x^2 - 1})(x + \sqrt{x^2 - 1})} = \frac{x^2 + x\sqrt{x^2 - 1}}{x^2 - (x^2 - 1)} = x^2 + x\sqrt{x^2 - 1}$$
So the equation becomes:
$x \frac{dy}{dx} + y = x^2 + x\sqrt{x^2 - 1}$.

Step 3: Integrate both sides.
Notice that the left-hand side is the derivative of $xy$:
$$\frac{d}{dx}(xy) = x^2 + x\sqrt{x^2 - 1}$$
Integrating with respect to $x$:
$xy = \int x^2 dx + \int x\sqrt{x^2 - 1} dx$
$xy = \frac{x^3}{3} + \frac{1}{3}(x^2 - 1)^{3/2} + C$.

Step 4: Use the initial condition $y(1) = 1$.
Substitute $x=1, y=1$:
$1(1) = \frac{1^3}{3} + \frac{1}{3}(1^2 - 1)^{3/2} + C$
$1 = \frac{1}{3} + 0 + C \implies C = \frac{2}{3}$.
So, $xy = \frac{x^3 + (x^2 - 1)^{3/2} + 2}{3}$.

Step 5: Evaluate $y(\sqrt{5})$.
Substitute $x = \sqrt{5}$:
$$\sqrt{5} y = \frac{(\sqrt{5})^3 + ((\sqrt{5})^2 - 1)^{3/2} + 2}{3}$$
$$\sqrt{5} y = \frac{5\sqrt{5} + (4)^{3/2} + 2}{3} = \frac{5\sqrt{5} + 8 + 2}{3} = \frac{5\sqrt{5} + 10}{3}$$
Divide by $\sqrt{5}$:
$y = \frac{5 + 2\sqrt{5}}{3}$.

Step 6: Find the greatest integer less than $y$.
Using $\sqrt{5} \approx 2.236$:
$y \approx \frac{5 + 2(2.236)}{3} = \frac{5 + 4.472}{3} = \frac{9.472}{3} \approx 3.1573$.
The greatest integer less than 3.1573 is 3.