The value of $${x_3}$$ obtained by solving the following system of linear equations is $$${x_1} + 2{x_2} - 2{x_3} = 4$$$ $$$2{x_1} + {x_2} + {x_3} = - 2$$$ $$$ - {x_1} + {x_2} - {x_3} = 2$$$

During the numerical solution of a first order differential equation using the Euler (also known as Euler Cauchy) method with step size $$h,$$ the local truncation error is of the order of

The solution of the differential equation $${{{d^2}y} \over {d{x^2}}} = 0$$ with boundary conditions
(i) $${{dy} \over {dx}} = 1$$ at $$x=0$$
(ii) $${{dy} \over {dx}} = 1$$ at $$x=1$$

The homogeneous part of the differential equation $$\,{{{d^2}y} \over {d{x^2}}} + p{{dy} \over {dx}} + qy = r\,\,$$ ( $$p, q, r$$ are constants) has real distinct roots if