Use the method of undetermined coefficients to determine only the form of a particular solution for the system: X'(t) = AX(t) + f(t), where A and f(t) are given.
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![Use the method of undetermined coefficients to determine only the form of a particular solution for the system:
X'(t) = AX(t) + f(t), where A and f(t) are given.
A = [ - 2 - 3 ] , f(t) = [ en ³²]
-6
OXp = sin 3t a + b OXp=sinta + tb O Xp = sin 3t a + cos 3t b + c O Xp = sinta + cost b+c
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- Write the given second order equation as its equivalent system of first order equations. u" + 4.5u' – 7.5u = -7 sin(3t), u(1) = -7.5, u'(1) = -6 Use v to represent the "velocity function", i.e. v = u'(t). Use v and u for the two functions, rather than u(t) and v(t). (The latter confuses webwork. Functions like sin(t) are ok.) u' = %3D Now write the system using matrices: u d + dt V V and the initial value for the vector valued function is: u(1) v(1)tion 1.7 For this system of two equations y = cos² (x), x² + y² = x = 2, = a. Plot the two equations and observe two intersections that occur near x 2 and x -1. b. Substitute y from the first equation into the second getting an equation in only x. Solve this equation for the x-value (s) by any of the methods in this chapte to find where the function is zero, then substitute this x-value in either of the original equations to ge y-values. - . Repeat part (b) but now solve for x from one equa tion and use it to eliminate x from the other. You should get the same solutions, of course, but which is easier to use?Take the system Find a particular solution of the form *₁(t) = x₂ (t) = help (formulas) = v cos(4t) 2 cos(4 #* - [73] - [5 ccm (d)] = 1 0
- Solve for the particular solution: (D2 - 5D + 6)y= 20sin(x) + 30cos(x) O yp = 5 sin(x) - 2 cos(x) O yp = -2 sin(x) + 2 cos(x) O yp = -2 sin(x) + 3 cos(x) yp = = sin(x) + 5 cos(x)5. Find the general solution of the system: x' - (: ) * -8°Find the values of a, b, c, and d when f (x) = ax³ + bx² + cx + d and the following conditions are true: a, b, c, & d are constants f (0) f(1) = 0 f'(1) = -1 f"(0) = -6 = 2 Hints: Start by taking the derivative of f You might have to solve a system of equations
- solve the system X' +y' + 2X = 0 Xg" -x-y: sint. (1) (2)HW: ans./ [-1, co) Solve 5- 3x (1) (2/3, co) (2) 1 2. X-1< -X- 2. (13/3, o ) 2. (3) Y- 31 y-5 2. (- co, 49/30] 4 [-4, co) (5) - Sx+- 4-The solution of the system X'= Select the correct answer. Seleccione una: a. b. e. 2-2 X is X=ce 0-2 [(i) cos(21)-(i)sin (21)] + c₂0-2¹[(1) cos(21) + (i)sin (28)] X=c₁e²4[(i) cos(2+)-(i) sin (21)] + c₂0² [(1) cos(28) + (i)sin (20)] c. none of these d. C20 X=c₁0² [(1) cost + (i) sint] + c₂e²-[(1) cost+ (i)sinr] X=c₁e-²-[(i) cost-(i)sint] + c₂e-2 [(1) cost + (i)sint]
- The functions y = x' +2x-3 and y=x-1 are graphed on the grid shown below. Which of the following is the solution set of the equation: I x+2x-3=x-1 (1) {-2, 1} (3) {-3, 0} (2) {-3, 1} (4) {-1, 4}Write the given third order linear equation as an equivalent system of first order equations with initial values. 2y" ty" y' sin(t) = cos(t) with y(-3) = 2, y'(-3) = 2, y'(-3) = −3 Use x₁ = y, x₂ = y', and x3 =y". = 0 0 - sin t with initial values 1 = 1 0 -t 2 2 -3 0 1 0 x + If you don't get this in 2 tries, you can get a hint. 0 0Use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t. x' = 9x - 10y + sint and y' = 5x - y - cost. Write the system in operator form.