List of top Mathematics Questions on Differential Equations asked in IISER

Which one of the following is the solution of the differential equation
\[ x^2 \frac{dy}{dx} + 9xy = x^4 \quad (\text{for } x > 0) \]
given that $y = 0$ when $x = 1$?

Consider the differential equation
$\cos(y) \frac{dy}{dx} + \frac{1}{x} \sin(y) = x, \quad (x > 0)$;
given that $y = \frac{\pi}{2}$ at $x = \sqrt{3}$. Which one of the following is the value of $y$ at $x = \sqrt{\frac{3}{2}}$ ?