We are given the differential equation: \[ \frac{dy}{dx} = \frac{y}{x}. \]
This is a separable differential equation, so we can rewrite it as: \[ \frac{dy}{y} = \frac{dx}{x}. \]
Now, integrating both sides: \[ \int \frac{1}{y} dy = \int \frac{1}{x} dx, \] \[ \ln |y| = \ln |x| + C. \]
Exponentiating both sides: \[ |y| = e^{\ln |x| + C} = |x| e^C. \] Thus, \( y = Cx \).
Using the initial condition \( y(1) = 2 \), we get: \[ 2 = C(1) \quad \Rightarrow \quad C = 2. \]
Therefore, the solution is \( y = 2x \).
Let one focus of the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ be at $ (\sqrt{10}, 0) $, and the corresponding directrix be $ x = \frac{\sqrt{10}}{2} $. If $ e $ and $ l $ are the eccentricity and the latus rectum respectively, then $ 9(e^2 + l) $ is equal to:
The largest $ n \in \mathbb{N} $ such that $ 3^n $ divides 50! is: