olve the following initial value problems for the systems of equations using the matrix method. Find eigenvalues and eigenvectors and fix the undetermined coefficients by using technology (for example, Matlab or WolframAlpha) (a) x′ = x + 2y + 2z y′ = 2x + 7y + z z′ = 2x + y + 7z , x(0) = 9, y(0) = 1, z(0) = −1. (b) x′ = 9x − 7y − 5z y′ = −12x + 7y + 11z + 9w z′ = 24x − 17y − 19z − 9w w′ = −18x + 13y + 17z + 9w , x(0) = 2, y(0) = 0, z(0) = 1, w(0) = 1

An aluminum wire having a cross-sectional area equal to 4.60 x 10-6 m? carries a current of 7.50 A. The density of aluminum is 2.70 g/cm³. Assume each aluminum atom supplies one conduction electron per atom. Find the drift speed of the electrons in the wire. 1.95E-4 The equation for the drift velocity includes the number of charge carriers per volume, which in this case is equal to the number of atoms per volume. How do you calculate that if you know the density and the atomic weight of aluminum? mm/s

Using MATLAB functions poly and polyder, compute the relative condition numbers of the roots 1, 2, ..., 20 of the Wilkinson polynomial 20 w(x) = (x1)(x-2)(x - 20) = x²0 210x¹⁹+... 19 with respect to perturbation of the coefficient of 2x¹⁹, a19, from -210 to -210+2-23 Present your results in a tabular form and write your conclusion on the ill-conditioninig of the Wilkinson polynomial. (Recall: condition number for the jth root is kj |a₁x²=1| \w'(x₂)

2. calculates the trajectory r(t) and stores the coordinates for time steps At as a nested list trajectory that contains [[xe, ye, ze], [x1, y1, z1], [x2, y2, z2], ...]. Start from time t = 0 and use a time step At = 0.01; the last data point in the trajectory should be the time when the oscillator "hits the ground", i.e., when z(t) ≤ 0; 3. stores the time for hitting the ground (i.e., the first time t when z(t) ≤ 0) in the variable t_contact and the corresponding positions in the variables x_contact, y_contact, and z_contact. Print t_contact = 1.430 X_contact = 0.755 y contact = -0.380 z_contact = (Output floating point numbers with 3 decimals using format (), e.g., "t_contact = {:.3f}" .format(t_contact).) The partial example output above is for ze = 10. 4. calculates the average x- and y-coordinates 1 y = Yi N where the x, y, are the x(t), y(t) in the trajectory and N is the number of data points that you calculated. Store the result as a list in the variable center = [x_avg, y_avg]…