The general solution of the differential equation $$\,\,{{{d^2}y} \over {d{x^2}}} + 2{{dy} \over {dx}} - 5y = 0\,\,\,$$ in terms of arbitrary constants $${K_1}$$ and $${K_2}$$ is

The residues of a function $$f\left( z \right) = {1 \over {\left( {z - 4} \right){{\left( {z + 1} \right)}^3}}}$$ are

An integral $${\rm I}$$ over a counter clock wise circle $$C$$ is given by $${\rm I} = \oint\limits_c {{{{z^2} - 1} \over {{z^2} + 1}}} \,\,{e^z}\,dz$$
If $$C$$ is defined as $$\left| z \right| = 3,$$ then the value of $${\rm I}$$ is

In the circuit shown, $$V$$ is a sinusoidal voltage source. The current $$I$$ is in phase with voltage $$V$$. The ratio $${{{\rm{Amplitude of voltage across the capacitor }}} \over {{\rm{Amplitude of voltage across the resistor }}}}$$ is ______.