In order to numerically solve the ordinary differential equation dy/dt = -y for t > 0, with an initial condition y(0) = 1, the following scheme is employed:

$\frac{y_{n+1} - y_{n}}{\Delta t} = -\frac{1}{2}(y_{n+1} + y_{n}).$

Here, $\Delta t$ is the time step and $y_n = y(n\Delta t)$ for $n = 0, 1, 2, \ldots.$ This numerical scheme will yield a solution with non-physical oscillations for $\Delta t > h.$ The value of h is

The value of the surface integral

where S is the external surface of the sphere x² + y² + z² = R² is

Let f(z) be an analytic function, where z = x + iy . If the real part of f(z) is cosh x cos y , and the imaginary part of f(z) is zero for y = 0 , then f(z) is

Consider the system of linear equations

x + 2y + z = 5

2x + ay + 4z = 12

2x + 4y + 6z = b

The values of a and b such that there exists a non-trivial null space and the system admits infinite solutions are