The equation \[ y'' + p(x)y' + q(x)y = r(x) \] is a _________ ordinary differential equation.
Let \[ A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & k & 0 \\ 3 & 0 & -1 \end{pmatrix}. \] If the eigenvalues of \( A \) are -2, 1, and 2, then the value of \( k \) is _. (Answer in integer)
Let \( a_0 = 0 \) and define \( a_n = \frac{1}{2} (1 + a_{n-1}) \) for all positive integers \( n \geq 1 \). The least value of \( n \) for which \( |1 - a_n|<\frac{1}{2^{10}} \) is ______.
The value of \( k \), for which the linear equations \( 2x + 3y = 6 \) and \( 4x + 6y = 3k \) have at least one solution, is ________. (Answer in integer)
Consider the matrix:
\[ A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix} \]
The eigenvalues of the matrix are 0.27 and ____ (rounded off to 2 decimal places).
\( \hat{i} \) and \( \hat{j} \) denote unit vectors in the \( x \) and \( y \) directions, respectively. The outward flux of the two-dimensional vector field \( \vec{v} = x \hat{i} + y \hat{j} \) over the unit circle centered at the origin is ___________ (rounded off to two decimal places).
An approximate solution of the equation \( x^3 - 17 = 0 \) is to be obtained using the Newton-Raphson method. If the initial guess is \( x_0 = 2 \), the value at the end of the first iteration is \( x_1 = \) ________ (rounded off to two decimal places).
Consider the ordinary differential equation:
The maximum value of the function \( f(x) = (x - 1)(x - 2)(x - 3) \) in the domain [0, 3] occurs at \( x = \) _________ (rounded off to two decimal places).
The partial differential equation \[ \frac{\partial^2 u}{\partial x^2} + 4 \frac{\partial^2 u}{\partial x \partial y} + 2 \frac{\partial^2 u}{\partial y^2} = 0 \] is ________.
