CSIR NET Maths previous year question papers

Let u = u(x, y) be the solution to the Cauchy problem $(y + u)\partial{}u/\partial{}x + y \partial{}u/\partial{}y = x - y, x \in \mathbb{R}, y > 0, u(x, 1) = 1 + x, x \in \mathbb{R}$. Then which of the following statements is true?

Let u = u(x, t) be a solution of the wave equation $\partial^{2}u/\partial{}t^{2} - \partial^{2}u/\partial{}x^{2} = 0, x \in \mathbb{R}, t > 0$ satisfying the condition u(0, t) = 0, ∀t ≥ 0. Then which of the following statements is true?

Let S denote the set of all solutions of the Euler-Lagrange equation of the variational problem: minimize $J[y] = \int_{0}^{1}(y^{2} + (y')^{2})$dx, subject to $y(0) = 0, y(1) = 0, \int_{0}^{1}y^{2}$dx = 1. Then the set {$\varphi(1/2) : \varphi \in S$} is equal to

Let B(0,1) = {$(x, y) \in \mathbb{R}^{2}$ | $x^{2} + y^{2} < 1$} be the open unit disc in $\mathbb{R}^{2}, \partial{}B(0,1)$ denote the boundary of B(0,1), and $\nu$ denote the unit outward normal to $\partial{}B(0,1)$. Let $f : \mathbb{R}^{2} \to \mathbb{R}$ be a given continuous function. The Euler–Lagrange equation of the minimization problem min { ½∬_B(0,1) |$\nabla{}u$|$^{2}$ dxdy + ½∬$_B(0,1) e^(u^{2})$ dxdy $+ \int_\partial{}B(0,1) f u$ ds } subject to $u \in C^{1}($closure of B(0,1)) is