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

Let \( y(x) \) be the solution of the differential equation \[ \frac{dy}{dx} = (y - 1)(y - 3) \] satisfying the condition \( y(0) = 2 \). Then which of the following is/are TRUE?

Let \( k, \ell \in \mathbb{R} \) be such that every solution of \[ \frac{d^2y}{dx^2} + 2k \frac{dy}{dx} + \ell y = 0 \] satisfies \( \lim_{x \to \infty} y(x) = 0 \). Then which of the following is/are TRUE?

Let \( y(x) \) be the solution of the differential equation \[ (xy + y + e^{-x}) \, dx + (x + e^{-x}) \, dy = 0 \] satisfying \( y(0) = 1 \). Then \( y(-1) \) is equal to

A particular integral of the differential equation \[ \frac{d^2y}{dx^2} - 2 \frac{dy}{dx} = e^{2x} \sin x \] is