(A) Determine the Solution of the Differential Equation $\frac { d y } { d x } = - y$

Essay

(a) Determine the solution of the differential equation $\frac { d y } { d x } = - y$ where $y ( 0 ) = 1$ .(b) Use the solution $y ( x )$ from part (a) to calculate $y ( 0.4 )$ .(c) Use Euler's Method with the given step sizes to estimate the value of $y ( 0.4 )$ for the equation given in part (a).(i) $h = 0.4$ (ii) $h = 0.2$ (iii) $h = 0.1$ (d) Sketch $y ( x )$ from part (b) and each of the Euler approximations from part (c) on the same set of axes.