CSIR NET Maths previous year question papers

Given that $y_{1}(x) = e^(2x)$ is a solution of the ordinary differential equation (ODE$) x d^{2}y/$dx$^{2} - (3 + 4x)$dy/dx + (4x + 6)y = 0, x > 0. Let $y_{2} = y_{2}(x)$ be the solution of the ODE satisfying the conditions $y_{2}(1) = e^{2}/4$ and (dy$_{2}/$dx$)(1) = 3e^{2}/2$. Then which of the following statements is true?

Let u = u(x, y) be the solution of the Cauchy problem $x \partial{}u/\partial{}x + y \partial{}u/\partial{}y = u, (x, y) \ne (0, 0)$, with $u(x, 1) = \sqrt{1 + x^{2}}, x \in \mathbb{R}$. 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 $f : \mathbb{R} \to \mathbb{R}$ be such that sup(x≠y) |f(x) − f(y)|/|x − y| = L, where $1 < L < \infty$. Let $h : \mathbb{R} \to \mathbb{R}$ be a differentiable function satisfying |h′(x)| ≤ 3/4 for all $x \in \mathbb{R}$. For $\alpha > 0$, define $g(x) = \alpha{}f(x) + h(x)$ for $x \in \mathbb{R}$. Consider the sequence {$x_{k}$}(k≥0) defined by $x_(k+1) = g(x_{k}), k = 0, 1, \dots$, where $x_{0} \in \mathbb{R}$. The sequence {$x_{k}$}(k≥0) converges to the solution of the equation x = g(x) if

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 $\lambda \in \mathbb{R}$, and $K : [0, 1] \times [0, 1] \to \mathbb{R}$ be a function such that every solution of the boundary value problem $(d^{2}u/$dx$^{2})(x) + \lambda{}u(x) = 0; ($du/dx)(0) = u(0), (du/dx)(1) = 0 satisfies the integral equation $u(x) + \lambda\int_{0}^{1}K(x, t)u(t)$dt = 0. Then

Two blocks of equal mass m are connected by a flexible inelastic cord of mass M. One block is placed on a smooth horizontal table, the other block hangs over the edge. The total potential energy of the entire cord is given by (−Mg$/2l)x^{2}$, where x is the distance of the hanging block from the edge of the table, l is the length of the cord, and g is the gravitational acceleration. Then

Consider the initial value problem (IVP$) y'(x) = \surd$|$y(x) + \varepsilon$|, $x \in \mathbb{R}, y(0) = y_{0}$. Consider the following statements: S1: There is an $\varepsilon > 0$ such that for all $y_{0} \in \mathbb{R}$, the IVP has more than one solution. S2: There is $a y_{0} \in \mathbb{R}$ such that for all $\varepsilon > 0$, the IVP has more than one solution. Then

Let $\varphi$ denote the solution to the boundary value problem (BVP) (xy′)′ − 2y′ + y/x = 1 for $1 < x < e^{4}$, with y(1) = 0 and $y(e^{4}) = 4e^{4}$. Then the value of $\varphi(e)$ is

Let u = u(x, t) be the solution of the following initial value problem: $u_{t} + 2024u_{x} = 0$ for $x \in \mathbb{R}, t > 0$, and $u(x, 0) = u_{0}(x)$ for $x \in \mathbb{R}$, where $u_{0} : \mathbb{R} \to \mathbb{R}$ is an arbitrary $C^{1}$ function. Consider the following statements: $S_{1}$: If $A_{t} :=$ {$x \in \mathbb{R} : u(x, t) < 1$} and |$A_{t}$| denotes the Lebesgue measure of $A_{t}$ for every t ≥ 0, then |$A_{t}$| = |$A_{0}$|, $\forall{}t > 0. S_{2}$: If $u_{0}$ is Lebesgue integrable, then for every t > 0, the function x ↦ u(x, t) is Lebesgue integrable. Then

If u = u(x, t) is the solution of the initial value problem $u_{t} = u_{xx}$ for $x \in \mathbb{R}, t > 0$, with u(x, 0) = sin(4x) + x + 1 for $x \in \mathbb{R}$, satisfying |u(x, t)| $< 3e^(x^{2})$ for all $x \in \mathbb{R}$ and t > 0, then

If the value of the approximate solution of the initial value problem $y'(x) = x(y(x) + 1), x \in \mathbb{R}, y(0) = \beta$ at x = 0.2 using the forward Euler method with step size 0.1 is 1.02, then the value of $\beta$ is

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

Let u be the solution of the Volterra integral equation $\int_{0}$ᵗ [½ $+ \sin(t - \tau)] u(\tau) d\tau = \sin t$. Then the value of u(1) is

Consider a solid circular cylinder of radius 2 meters and height 3 meters of uniform density. If the density of the cylinder is $\rho$ kg/meter$^{3}$, then the moment of inertia (in kg meter$^{2})$ of the cylinder about a diameter of its base is

Suppose x(t) is the solution of the initial value problem in $\mathbb{R}^{2}$: ẋ = Ax, $x(0) = x_{0}$, where A = [[5, 4], [1, 2]]. Which of the following statements is true?

Let u(x, y) be the solution of the Cauchy problem $u\cdot{}u_{x} + u_y = 0$ for $x \in \mathbb{R}, y > 0$, with u(x, 0) = x for $x \in \mathbb{R}$. Which of the following is the value of u(2, 3)?

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?

For the unknown $y : [0,1] \to \mathbb{R}$, consider the boundary value problem y″(x) + 2y(x) = 0 for x ∈ (0,1), y(0) = y(1) = 0. It is given that it corresponds to the integral equation $y(x) = 2\int_{0}^{1} K(x, t) y(t)$ dt. Which of the following is the kernel K(x, t)?

The smallest real number $\lambda$ for which the problem $-y'' + 3y = \lambda{}y, y(0) = 0, y(\pi) = 0$ has a non-trivial solution is

The partial differential equation $x^{2} u_{xx} - 2$xy $u_{x}$ᵧ $- 3y^{2} u_yy + u_{x} - u_y = 0$ is

Consider the Cauchy problem for the wave equation $u_tt - 4u_{xx} = 0$ on $\mathbb{R} \times (0, \infty)$, with u(x, 0) = $e^{-1/x^{2}}$ for x ≠ 0 (and 0 at x = 0), and u_t(x, 0) = x $e^{-x^{2}}$. Which one of the following is true?

Using Euler's method with step size 0.05, the approximate value of the solution of dy/dx $= \sqrt{3x + 2y + 1}, y(1) = 1$, at x = 1.1 (rounded to two decimal places) is

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

The value of $\lambda$ for which the integral equation $y(x) = \lambda\int_{0}^{1} x^{2}$ $e^{x+t}$ y(t) dt has a non-zero solution is