CSIR NET Maths previous year question papers

If $\varphi(x) = x$ is a solution of the ordinary differential equation (ODE$) d^{2}y/$dx$^{2} - (2/x^{2} + 1/x)(x$ dy/dx $- y) = 0, 0 < x < \infty$, then the general solution of the ODE is given by

Let $(\lambda_{n})_{n}\in\mathbb{N}$ be the sequence of eigenvalues of the Sturm-Liouville problem (d/dx)(x dy/dx$) + (\lambda/x)y = 0, 1 < x <$ $e^{2\pi}$, y(1) = 0, $y(e^{2\pi})$ = 0. Then $\sum_{n}$₌$_{1}^\infty 1/\lambda_{n}$ is equal to

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 the solution of $\partial^{2}u/\partial{}t^{2} - \partial^{2}u/\partial{}x^{2} = 0, x \in \mathbb{R}, t > 0, u(x, 0) = 1 + x^{2}, x \in \mathbb{R}, \partial{}u/\partial{}t(x, 0) = x + 1, x \in \mathbb{R}$. Then the value of u(1, 1) is

If the function $s : [0, 4] \to \mathbb{R}$ defined by $s(x) = a(x - 2)^{2} + b(x - 1)^{2}$ for $0 \le x \le 1, s(x) = (x - 2)^{2}$ for 1 < x ≤ 3, and $s(x) = 2c(x - 2)^{2} + (x - 3)^{3}$ for 3 < x ≤ 4, is a cubic spline, then the value of 2a + b + 2c is

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

If y(x) is the solution of the integral equation $y(x) = x^{2} + 2\int_{0}^{1}$ xt y(t) dt, then which of the following statements is true?

Suppose a dynamical system has the Lagrangian L = (q̇$_{1})^{2} + (q$̇$_{2})^{2} + (q_{1})^{2} + q$̇$_{1}q$̇$_{2}$. If $p_{1}$ and $p_{2}$ are momenta conjugate to $q_{1}$ and $q_{2}$ respectively, then which of the following statements is true?

Let u(x,t) be the solution of the partial differential equation $u_{t} = 16u_{xx} + 2, 0 < x < 7, t > 0$, satisfying the conditions $u_{x}(0,t) = u(7,t) = 0$ for t > 0, and u(x,0) = 0 for 0 < x < 7. Then which of the following statements is true?

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?

Consider the following statements: (P) The initial value problem y′ = f(y), where f(y) = y·cos(1/y) for y ≠ 0 and f(0) = 0, y(0) = 0, has at most one solution. (Q) If z(x) is the solution of the initial value problem $z' = (1-z^{4})^{100}/(1+z^{2}), z(0) = 0$, then for each $\alpha \in (0,2)$, there exists $x_\alpha \in \mathbb{R}$ such that $z(x_\alpha) = 1-\alpha$. Then which of the following statements is true?

Consider the initial-boundary value problem (IBVP$) u_{t} + u_{x} = 0, 0 < x < \infty, t > 0, u(x,0) = e$ˣ, $0 < x < \infty, u(0,t) = 1 - \sin t, t > 0$. Then which of the following statements is true?

For $a \in \mathbb{R}$, let $y_{1}(x)$ and $y_{2}(x)$ be solutions of the differential equation $y'' + (e^(x^{2}) + \cos x)y = 0$ such that $y_{1}(0) = 3, y_{1}'(0) = -1, y_{2}(0) = -5, y_{2}'(0) = a$. Suppose $W(y_{1},y_{2})(x)$ denotes the Wronskian of $y_{1}$ and $y_{2}$. If $W(y_{1},y_{2})(1/2) = 4$, then the value of a is

Let u(x,t) be the solution of the initial value problem $u_{tt} - u_{xx} = e^(-t), x \in \mathbb{R}, t > 0, u(x,0) = \cos x, u_{t}(x,0) = 0, x \in \mathbb{R}$. Then which of the following statements is true?

If p(x) is the interpolating polynomial for the data x = −2,−1,0,1,2 with y = −1,3,1,−1,3, then the value of p(1/2) is

Suppose u(x) is the solution of the integral equation $u(x) = 3 + \int_{0}$ˣ (x−t)u(t) dt. Then which of the following statements is true?

Consider the ordinary differential equation (ODE$) d^{2}y/$dx$^{2} + \cos(x)$ dy/dx + sin(x) y = 0. Let $\varphi_{1}(x), \varphi_{2}(x)$ be solutions of the ODE, satisfying $\varphi_{1}(0) = 1, d\varphi_{1}/$dx(0) = 0, and $\varphi_{2}(0) = 0, d\varphi_{2}/$dx(0) = 1. Then which of the following statements are true?

For a continuous function q defined on $\mathbb{R}$, consider the ordinary differential equation (ODE$) d^{2}y/$dx$^{2} + q(x)y = 0, x \in \mathbb{R}$. Then which of the following statements are FALSE?

Consider the Cauchy problem (CP$) x \partial{}u/\partial{}x + \partial{}u/\partial{}y = 1, u(0, y) = e^y, y \in \mathbb{R}$. Then which of the following statements are true?

Suppose u = u(x, y) is the solution of the boundary value problem $\partial^{2}u/\partial{}x^{2} + \partial^{2}u/\partial{}y^{2} + \partial{}u/\partial{}x = 0$ in {$(x, y) \in \mathbb{R}^{2} : x^{2} + y^{2} < 1$}, $u(x, y) = 1 + 2x^{2}y^{2}$ on {$(x, y) \in \mathbb{R}^{2} : x^{2} + y^{2} = 1$}. Then which of the following statements are true?

If $\alpha, \beta \in \mathbb{R}$ are such that the equation $\int_{0}^{3} f(x)$ dx $= (3/2)[f(\alpha) + f(\alpha + \beta)]$ holds for all polynomials f(x) of degree less than or equal to 2, then which of the following statements are 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?

Let D = {$(x, y) \in \mathbb{R}^{2} : -1 \le x \le 1, -1 \le y \le 1$}, and $f : D \to \mathbb{R}$ be the function defined by $f(x, y) = 1 + \surd(y$₊), where y₊ = max{y, 0}. Consider the initial value problem (IVP) dy/dx = f(x, y), y(0) = 0. Then which of the following statements are true?

Consider the initial value problem (IVP) y′ + y = 0, y(0) = 1. Let $(y_{n})$ be the iterates of forward Euler method, applied to the IVP, with step size h where 0 < h < 1. Then which of the following statements are true?