Differential Equations

3. Let A and B denote n x n matrices. For each of the following, prove the statement in general or give a counterexample that shows it is not true. (a) (A + B) (AB) = A²-B² (b) (I + A)² = 1 + 2A + A²

17. For each function f, find: a. fx b. fy C. fxy d. fyx f(x, y) = 2x5 - 3x²y³ + y4 − 3x + 2y +7

18. Sketch the graph of the following function. If you use a graphing calculator, make a hand-drawn sketch showing the amplitude and period.) y = 7cos2x

19. Find the derivative of the following function: f(t) = sin(2-3t)

4) Determine whether the following systems are causal: (a) y(t) = x(-t) and (b) y(t) = x(t+1)

6) Determine whether the following systems are BIBO-stable: (a) y(t) = x²(t) and (b) y(t) = tx(t)

Determine the 'utions for the following equations: 1). 2) 3). 0:1 م (x *(x² + y 2) + y) dx + (y *(x² + y²) + x) dy=0 2 (x y + x) y'=y y' + y/(1-x) = x²

4. Determine whether the set S = {-2t+t²,8+t³, -t² + t³, −4+ t²} spans P3.

Let A 4 be the R-vector space V = {p € R[x] / degree (p) 3). (a) Show that B = ((1+x)°, (1+x)1, (1+x)2, (1 + x)3) an ordered basis of V shall. What is the dimension of V? (b) Calculate the coordinate vector of p = 2x3 + 5x2 + 4x - 1 with respect to the Base B. (3 + 2 Points)

2. Two chemicals are mixed together and as they react the temperature of the mixture is measured. One, two and three hours after the mixing takes place, the temperatures of the chemicals are, respectively, 156.8, 161.7 and 177.2 degrees. (a) Create a system of linear equations for the data to fit the curve T(t) = at² + bt+c, where t is the time in hours and T' is the temperature. You essentially need to find a parabola that passes through the three points (1, 156.8), (2, 161.7), (3, 177.2). (b) Write the system in the form Ax = b. (c) Use Cramer's Rule to solve the system. Be sure to at least show the set up for using Cramer's Rule. You can use a computer to compute the determinants if so desired. (d) Write the equation of the parabola. 1 le