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 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

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?

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?

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?

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?

Let u(x,t) be the solution of the initial-boundary value problem $u_{tt} - u_{xx} + u_{t} = 0, 0<x<\pi, t>0, u(0,t)=0, u(\pi,t)=0, t>0, u(x,0)=\sin x, u_{t}(x,0)=0, 0<x<\pi$. For t≥0, define $E(t) = \int_{0}^\pi (u_{t}^{2}+u_{x}^{2})$ dx. Then which of the following statements are true?

Consider the boundary value problem (BVP$) u_{xx}+u_yy=0$ in $\Omega=${$(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}<1$}, u(x,y)=e^(x+y) on $\partial\Omega=${$(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}=1$}. Then which of the following statements are true?

The problem $\partial^{2}u/\partial{}x^{2} + \partial^{2}u/\partial{}y^{2} = 0$ in {$(x, y) \in \mathbb{R}^{2} : x^{2} + y^{2} > 1$}, u(x, y) = 1 on {$(x, y) \in \mathbb{R}^{2} : x^{2} + y^{2} = 1$} has

The problem $\partial{}u/\partial{}x + \partial{}u/\partial{}y = u^{2}, u(x, 0) = x^{2}, \forall{}x \in \mathbb{R}$ has a solution on an open set containing the line {$(x, y) \in \mathbb{R}^{2}$ : ax + by = 0} if

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 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

Let u = u(x, t) be the solution to the initial value problem $\partial^{2}u/\partial{}t^{2} - \partial^{2}u/\partial{}x^{2} = 0, x \in \mathbb{R}, t > 0, u(x, 0) = f(x), x \in \mathbb{R}, (\partial{}u/\partial{}t)(x, 0) = 1/\sqrt{1 + x^{2}}, x \in \mathbb{R}$, where f is a twice continuously differentiable function defined on $\mathbb{R}$ satisfying f(x) → 0 as |x| $\to \infty$. Then which of the following statements are true?

Consider the partial differential equation $4(\partial^{2}u/\partial{}x^{2}) + 6(\partial^{2}u/\partial{}x\partial{}y) + 3(\partial^{2}u/\partial{}y^{2}) = 0$. Which of the following partial differential equations can be obtained by a change of independent variables given by $(\xi, \eta)$ᵗ = A(x, y)ᵗ for some 2 × 2 invertible matrix A with real entries?

Let u = u(x, y) be the solution of the boundary value problem $\partial^{2}u/\partial{}x^{2} + \partial^{2}u/\partial{}y^{2} = 0$ on (0, 1) × (0, 1), with $u(x, 0) = e^(\pi{}x), u(x, 1) = -e^(\pi{}x)$ for $x \in [0, 1], u(0, y) = \cos(\pi{}y) + \sin(\pi{}y)$ and $u(1, y) = e^\pi(\cos(\pi{}y) + \sin(\pi{}y))$ for y ∈ [0, 1]. Then there exists a point $(x_{0}, y_{0}) \in (0, 1) \times (0, 1)$ such that

Let u = u(x, t) be the solution of the initial-boundary value problem $\partial{}u/\partial{}t = \partial^{2}u/\partial{}x^{2}$ on $(0, 1) \times (0, \infty)$, with u(x, 0) = 4x(1 − x) for x ∈ [0, 1] and u(0, t) = u(1, t) = 0 for t ≥ 0. Then which of the following statements are true?

Consider the initial boundary value problem (IBVP$) u_{t} + u_{x} = 2u$ for x > 0, t > 0; u(0, t) = 1 + sin t for t > 0; u(x, 0) = eˣ cos x for x > 0. If u is the solution of the IBVP, then the value of $u(2\pi, \pi)/u(\pi, 2\pi)$ is

Let B(0,2) = {$(x, y) \in \mathbb{R}^{2} : x^{2} + y^{2} < 4$}, and $\partial{}B$ denote the boundary of B(0,2). Assume $(\alpha, \beta) \ne (0,0), k \in \mathbb{R}$, and u is any solution to $-\Delta{}u = 0$ in $B(0,2), \alpha{}u(x, y) + \beta(\partial{}u/\partial\nu)(x, y) = 1 + (x^{2} + y^{2})k$ on $\partial{}B$, where $\nu(x, y)$ is the unit outward normal to B(0,2) at $(x, y) \in \partial{}B$. Consider the following statements: $S_{1}$: If $\beta = 0$, then there exists $a (x_{0}, y_{0}) \in B(0,2)$ such that |$u(x_{0}, y_{0})$| = |1 + 4k|/|$\alpha$|. $S_{2}$: If $\alpha = 0$, then k = −1/4. Then

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)?

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?

Let $u : \mathbb{R}^{2} \to \mathbb{R}$ solve $\partial_{x}u + 2\partial_yu = 0$ on $\mathbb{R}^{2}$ with u(x, y) = sin x on the line y = 3x + 1, and let $v : \mathbb{R}^{2} \to \mathbb{R}$ solve $\partial_{x}v + 2\partial_yv = 0$ with v(x, 0) = sin x. Let S = [0,1] × [0,1]. Which of the following statements are true?