Solution, the variable $${x_1}$$ and $${x_2}$$ for the following equations is to be obtained by employing the Newton $$-$$ Raphson iteration method
equation (i) $$10\,{x_2}\,\sin \,{x_1} - 0.8 = 0$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$10\,x_2^2\, - 10\,{x_2}\cos \,{x_1} - 0.6 = 0$$
Assuming the initial values $${x_1} = 0.0$$ and $${x_2} = 1.0$$ the Jacobian matrix is

A differential equation $${{dx} \over {dt}} = {e^{ - 2t}}\,\,u\left( t \right)\,\,$$ has to be solved using trapezoidal rule of integration with a step size $$h=0.01$$ sec. Function $$u(t)$$ indicates a unit step function. If $$x(0)=0$$ then the value of $$x$$ at $$t=0.01$$ sec will be given by

The value of $$\,\,\,\int\limits_1^2 {{1 \over x}\,\,\,dx\,\,\,\,} $$ computed using simpson's rule with a step size of $$h=0.25$$ is