Differential Equations · Mathematics · IAT (IISER)

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

Which of the following differential equations has $y=e^x$ as one of its particular solutions?

Consider the differential equation $f^{\prime \prime}+p f^{\prime}+q f=0$ where $p$ and $q$ are constants, $p^2=4 q$ and $q \neq 0$. If y is a nonzero solution of the equation, then which of the following statements is correct?