MATH.2360 Engineering Differential Equations Some Useful MATLAB Commands for Symbol Manipulation Type the following command in the command window, which will open a new window mupad notebook: called a >>mupad A. Algebra 1. MATLAB will factor polynomials. Type the following command in the mupad notebook next to the [ symbol, then hit the Enter key factor(x 4 5*x2 4) 2. MATLAB can solve single equations. Try solve (x^4-5*x^2 + 4 0) = 3. MATLAB can also solve systems of equations: solve (t2*x y-5, x + 2*y -4)) 4. MATLAB can simplify expressions. iand co(a)co) sin() Try these commands to simplify simplify(x/(2*x + 1) + 1/x) simplify ((cos (x))-3 cos (x)*(sin(x))-2. _+ - 2r+1 r B. Calculus MATLAB can find limits, derivatives, and integrals symbolically. Try the following commands to find lino sinrn. Inn e-r%PlA.JaT / dz, and dz, respectively limit(sin(x)/x, x-0) limit(exp(-x), x-infinity) diff (x 3, x) diff (x 3, x$2) int (1/(x 2+1), x) int(x/(x 2+1)~(3/2), x) OVER C. Differential Equations MATLAB can even solve differential equations symbolically. Here are some examples: 1. To solve the d.e. ay+2ry 32, type the command solve (ode (x"2*y' (x) + 2*x*y(x) -3sx"2, y(x))) You can also solve initial value problems, such as y, = y, y(1) = 1: solve (ode({y' (x) = y(x)^2, 2. y(1) -1), y(x))) 3. You can name the solution of an initial value problem, and you can even calculate the value of the solution at any r. First enter this command: y:=solve(ode ({y',(x) -y(x) = 0, y(0) = 1, and then enter this command float (yi I x-2) y, (0) 2}, y(x))) = D. Exercises (These are from the homework for section 1.5.,) I. Solve the initial value problem y, + y = 2, y(0) = 0. 2. Solve the initial value problem xy-y-x, y(1)-7