List of top Mathematics Questions on Differential Equations asked in IIT JAM MA

Let \(\alpha, \beta\) and \(\gamma\) be fixed real numbers such that

Let \(\alpha\) and \(\beta\) be real numbers such that the differential equation

Consider the differential equation \[ (2y \cos x - x y \sin x) \, dx + 2x \cos x \, dy = 0 \quad \text{for} \, x \in (0, \frac{\pi}{4}). \]
Which of the following is/are integrating factor(s) of the differential equation?

Given that the solution of \[ \frac{d^2y}{dx^2} + \alpha \frac{dy}{dx} + \beta y = -e^{-x} \] is \[ y(x) = C_1 e^{-x} + C_2 e^{2x} + x e^{-x}, \] find the values of $\alpha$ and $\beta$.