List of top Mathematics Questions on types of differential equations asked in JEE Advanced

Let $y = f(x)$ be the real valued function defined on the interval $(0, \infty)$, satisfying $f(1) = 0$ and the differential equation \[ x \frac{dy}{dx} = y - x^3. \] Then which of the following statements is (are) TRUE?
Let \(y : (-\infty, \infty) \to (0, \infty)\) be the solution of the differential equation \[ \frac{dy}{dx} = \frac{e^{5x}y^3 + y^3}{e^x + e^x y^4} \] satisfying \[ y(0) = \frac{1}{\sqrt{2}}. \] Then the value of \[ y(\log_e 2) \] is: