The solution of the initial value problem $$\,\,{{dy} \over {dx}} = - 2xy;y\left( 0 \right) = 2\,\,\,$$ is

Consider an ordinary differential equation $${{dx} \over {dt}} = 4t + 4.\,\,$$ If $$x = {x_0}$$ at $$t=0,$$ the increment in $$x$$ calculated using Runge-Kutta fourth order multi-step method with a step size of $$\Delta t = 0.2$$ is

If $$z$$ is a complex variable, the value of $$\int\limits_5^{3i} {{{dz} \over z}} $$ is

Laplace transform of $$\cos \,\left( {\omega t} \right)$$ is $${s \over {{s^2} + {\omega ^2}.}}$$. The Laplace transform of $${e^{ - 2t}}\,\cos \left( {4t} \right)$$ is