CSIR NET Maths previous year question papers

Let y(x) be the extremal of the functional $J[y] = \int_{0}^{\pi/4}$ $((y')^{2} - 4y^{2} + 2$xy) dx subject to $y(0) = 0, y(\pi/4) = 1$. Then y(x) is equal to

Suppose y(x) is the extremal of the variational problem $J(y) = \int_{0}^{4} (y')^{2}/y^{2}$ dx subject to $y(0) = 1, y(4) = e^{8}$. Then which of the following statements is true?

For $\alpha \ge 0$, consider the functional $J_\alpha[y] = \int_{1}^{2} (y')^{2}/x^\alpha$ dx defined for all continuously differentiable functions defined on the interval [1, 2] satisfying the conditions y(1) = 1, y(2) = 2. Then which of the following statements are true?

For $a, b, c \in \mathbb{R}$, consider the variational problem: Minimize $J[y] = \int_{0}^{2} [a(y')^{2} + 2$byy′ + cy$^{2}]$ dx, subject to y(0) = 10, y(1) = 100. Then which of the following statements are true?

Suppose y(x) is the extremal of the variational problem $J(y) = \int_{0}^{1} x^{2}(y')^{2}$ dx subject to $y(0)=0, y(1)=1, \int_{0}^{1}y^{2}$dx = 1/7. Then which of the following statements are true?

Suppose y(x) is the extremal of the variational problem $J(y) = \int_{0}^{1} ((y')^{2}\sin x + (2cos x)y)$ dx subject to y(0)=0, y(1)=1. Then which of the following statements are true?

If $\varphi \in C^{4}[0, 1]$ is the extremal of the variational problem minimize $J[y] = \int_{0}^{1}($yy$' + (y'')^{2})$dx, subject to y(0) = 0, y′(0) = 1, y(1) = 2, y′(1) = 4, then $\varphi(1/2)$ is equal to

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

Define S := {$y \in C^{1}[0, 1] : y(0) = y(1) = 0$}, ‖y‖$(\infty) := \max(x\in[0,1])$|y(x)| for every $y \in S, B_{0}(0, \varepsilon) :=$ {y ∈ S : ‖y‖$(\infty) < \varepsilon$}, $B_{1}(0, \varepsilon) :=$ {y ∈ S : ‖y‖$(\infty) +$ ‖y′‖$(\infty) < \varepsilon$}. Consider the functional $J : S \to \mathbb{R}$ given by $J[y] = \int_{0}^{1}[(y')^{2} - 2x(y')^{4}]$dx, then there exists an $\varepsilon > 0$ such that

Consider the variational problem minimize $J[y] = \int_{0}^{1}[(y')^{2} + 2025$ yy′]dx, where y(0) and y(1) are free. Then which of the following statements are true?

For any $b \in \mathbb{R}$, let S(b) denote the set of all broken extremals with one corner of the variational problem: minimize $J[y] = \int_{0}^{1}((y')^{4} - 3(y')^{2})$dx, subject to y(0) = 0, y(1) = b. Then which of the following statements are true?

Define S := {$y \in C^{1}[-1, 1] : y(-1) = -1, y(1) = 3$}. Let $\varphi$ be the extremal of the functional $J : S \to \mathbb{R}$ given by $J[y] = \int$₋$_{1}^{1}[(y')^{3} + (y')^{2}]$dx. Define ‖y‖$(\infty) := \max(x\in[-1,1])$|y(x)| for every y ∈ S and let $B_{0}(\varphi, \varepsilon) :=$ {y ∈ S : ‖$y - \varphi$‖$(\infty) < \varepsilon$}, $B_{1}(\varphi, \varepsilon) :=$ {y ∈ S : ‖$y - \varphi$‖$(\infty) +$ ‖$y' - \varphi'$‖$(\infty) < \varepsilon$}. Then which of the following statements are true?

The infimum of the set {$\int_{a}$ᵇ $\sqrt{1 + (y'(t))^{2}}$dt : $y \in C^{1}[a, b], y(a) = a^{2}, y(b) = b - 5$} is

The extremizer of the problem $\min[(1/2)\int$₋$_{1}^{1} ((y'(x))^{2} + (y(x))^{2})$dx] subject to $y \in C^{1}[-1, 1], \int$₋$_{1}^{1}$ xy(x)dx = 0 and y(−1) = y(1) = 1 is

Consider the variational problem $(P): J(y) = \int_{0}^{1} [(y')^{2} - y$|y|·y′ + xy] dx, y(0) = 0, y(1) = 0. Which of the following statements is correct?

The cardinality of the set of extremals of $J[y] = \int_{0}^{1} (y')^{2}$ dx subject to $y(0) = 1, y(1) = 6, \int_{0}^{1} y$ dx = 3 is

Let y(x) and z(x) be the stationary functions (extremals) of $J(y, z) = \int_{0}^{1} [(y')^{2} + (z')^{2} + y'z']$ dx subject to y(0) = 1, y(1) = 0, z(0) = −1, z(1) = 2. Which of the following statements are correct?

Among the curves connecting the points (1, 2) and (2, 8), let $\gamma$ be the curve on which an extremal of $J[y] = \int_{1}^{2} (1 + x^{3}y') y'$ dx can be attained. Which of the following points lie on the curve $\gamma$?

Define S = {$y \in C^{1}[0, \pi] : y(0) = y(\pi) = 0$}, ‖f‖$_\infty = \max$ |f|, $B_{0}(0, \varepsilon) =$ {f ∈ S : ‖f‖$_\infty < \varepsilon$} and $B_{1}(0, \varepsilon) =$ {f ∈ S : ‖f‖$_\infty +$ ‖f′‖$_\infty < \varepsilon$}. For the functional $J[y] = \int_{0}^\pi (1 - (y')^{2}) y^{2}$ dx, there exists $\varepsilon > 0$ such that