A numerical solution of the equation $$f\left( x \right) = x + \sqrt x - 3 = 0$$ can be obtained using Newton $$-$$ Raphson method. If the starting values is $$x=2$$ for the iteration then the value of $$x$$ that is to be used in the next step is

Consider a differential equation $${{dy\left( x \right)} \over {dx}} - y\left( x \right) = x\,\,$$ with initial condition $$y(0)=0.$$ Using Euler's first order method with a step size of $$0.1$$ then the value of $$y$$ $$(0.3)$$ is

The recursion relation to solve $$x = {e^{ - x}}$$ using Newton $$-$$ Raphson method is

The equation $${x^3} - {x^2} + 4x - 4 = 0\,\,$$ is to be solved using the Newton - Raphson method. If $$x=2$$ taken as the initial approximation of the solution then the next approximation using this method, will be