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dy = 9. Rewrite the given differential equation...
For the system described by the following differential equation d3y(t) d2y(t) d2x(t) dy(t) 3 dt dx(t)...
For the system described by the following differential equation d3y(t) d2y(t) d2x(t) dy(t) 3 dt dx(t) 9 dt y(t) 5x(t) 7 2 6 dt3 dt2 dt2 Express the system transfer function using the pole-zero plot technique a) b) What can be said about the stability of this stem?
For the system described by the following differential equation d3y(t) d2y(t) d2x(t) dy(t) 3 dt dx(t) 9 dt y(t) 5x(t) 7 2 6 dt3 dt2 dt2 Express the system transfer function using...
Find the state equations for the following differential equation: dy(t) dy(t) + 6y(t) = z(t) dt3...
Find the state equations for the following differential equation: dy(t) dy(t) + 6y(t) = z(t) dt3 dt2 dt +60360) + 11. 1 1 + oz(t) 0 0 -6 -11 X3 X 1 y(t) = [1 0 0] x2 + [0]z(t) X3 X1 0 1 0 = 0 0 0 X2 0 Iz(t) X2 6 11 6 X3 X3 X1 y(t) = [1 1 0] x2 + [O] z (t) X3
Find the general solution to the homogeneous differential equation dạy dt2 229 dy dt + 117y...
Find the general solution to the homogeneous differential equation dạy dt2 229 dy dt + 117y = 0 The solution can be written in the form y = Cjepit + Czert with ri < r2 Using this form, r1 = and r2 = BE SURE TO WRITE THE SMALLER r FIRST!
Problem 4. The higher order differential equation and initial conditions are shown as follows: = dy...
Problem 4. The higher order differential equation and initial conditions are shown as follows: = dy dy +y?, y(0) = 1, y'(0) = -1, "(0) = 2 dt3 dt (a) [5pts. Transform the above initial value problem into an equivalent first order differential system, including initial conditions. (b) [2pts.] Express the system and the initial condition in (a) in vector form. (c) [4pts.] Using the second order Runge Kutta method as follows Ū* = Ūi + hĚ(ti, Ūi) h =...
Use the substitution x = et to transform the given Cauchy-Euler equation to a differential equation...
Use the substitution
x =
et
to transform the given Cauchy-Euler equation to a differential
equation with constant coefficients. (Use yp for
dy
dt
and ypp for
d2y
dt2
.)
x2y'' +
10xy' + 8y =
x2
Solve the original equation by solving the new equation using the
procedures in Sections 4.3-4.5.
y(x) =
Use the substitution x = ef to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. (Use yp for dy and ypp for...
Given the following differential equation for some plant, dy +7.+ 15y = 2x(t) dt dt a....
Given the following differential equation for some plant, dy +7.+ 15y = 2x(t) dt dt a. Find the steady-state output for a unit-step input. b. Find the step response of the plant; that is, solve for the output if the input is a step function, x(t) = u(t).
Consider the system of two coupled differential equations: y-cx + dy, x-ax + by, with the equilibrium solution (xe,ye) = (0,0) (a) Rewrite the coupled system as a matrix differential equation and ide...
Consider the system of two coupled differential equations: y-cx + dy, x-ax + by, with the equilibrium solution (xe,ye) = (0,0) (a) Rewrite the coupled system as a matrix differential equation and identify the matrix A. Obtain a general solution to the matrix differential equation in terms of eigenvectors and eigenvalues of A. Justify your answer (b) Classify possible types and stability of the equilibrium with dependence on the eigenvalues of A. (Note: You are not asked to compute the...
Express the system of differential equations in matrix notation x – 4x + y - (cos...
Express the system of differential equations in matrix notation x – 4x + y - (cos t)x = 0 y"+y" - t?x' + 3y'+e-2x = 0 Which of the following sets of definitions allows the given system to be written as an equivalent system in normal form using only the new variables? OA. Xi =X, X2 = X". X3 = y, Xa =y" O B. *= x, X2 = x', *3 = y, X4 =y', X5 =y" OC. *1 =...
Solve the following differential equation using variation of parameters. d yt) 2 dy() +7- + 10y() u(t) dt dt2 y(0) 0...
Solve the following differential equation using
variation of parameters.
d yt) 2 dy() +7- + 10y() u(t) dt dt2 y(0) 0, y'(0) = 3
d yt) 2 dy() +7- + 10y() u(t) dt dt2 y(0) 0, y'(0) = 3
Along with x1' please solve for x2'. Thanks! Transform the given differential equation into an equivalent...
Along with x1' please solve for
x2'. Thanks!
Transform the given differential equation into an equivalent system of first-order differential equations. y' (t) + 5y' (t) - 6ty(t) = 6 cost Let x, = y and X, Ey. Complete the differential equation for X.
For the system described by the following differential equation d3y(t) d2y(t) d2x(t) dy(t) 3 dt dx(t) 9 dt y(t) 5x(t) 7 2 6 dt3 dt2 dt2 Express the system transfer function using the pole-zero plot technique a) b) What can be said about the stability of this stem?
For the system described by the following differential equation d3y(t) d2y(t) d2x(t) dy(t) 3 dt dx(t) 9 dt y(t) 5x(t) 7 2 6 dt3 dt2 dt2 Express the system transfer function using...
Find the state equations for the following differential equation: dy(t) dy(t) + 6y(t) = z(t) dt3 dt2 dt +60360) + 11. 1 1 + oz(t) 0 0 -6 -11 X3 X 1 y(t) = [1 0 0] x2 + [0]z(t) X3 X1 0 1 0 = 0 0 0 X2 0 Iz(t) X2 6 11 6 X3 X3 X1 y(t) = [1 1 0] x2 + [O] z (t) X3
Find the general solution to the homogeneous differential equation dạy dt2 229 dy dt + 117y = 0 The solution can be written in the form y = Cjepit + Czert with ri < r2 Using this form, r1 = and r2 = BE SURE TO WRITE THE SMALLER r FIRST!
Problem 4. The higher order differential equation and initial conditions are shown as follows: = dy dy +y?, y(0) = 1, y'(0) = -1, "(0) = 2 dt3 dt (a) [5pts. Transform the above initial value problem into an equivalent first order differential system, including initial conditions. (b) [2pts.] Express the system and the initial condition in (a) in vector form. (c) [4pts.] Using the second order Runge Kutta method as follows Ū* = Ūi + hĚ(ti, Ūi) h =...
Use the substitution
x =
et
to transform the given Cauchy-Euler equation to a differential
equation with constant coefficients. (Use yp for
dy
dt
and ypp for
d2y
dt2
.)
x2y'' +
10xy' + 8y =
x2
Solve the original equation by solving the new equation using the
procedures in Sections 4.3-4.5.
y(x) =
Use the substitution x = ef to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. (Use yp for dy and ypp for...
Given the following differential equation for some plant, dy +7.+ 15y = 2x(t) dt dt a. Find the steady-state output for a unit-step input. b. Find the step response of the plant; that is, solve for the output if the input is a step function, x(t) = u(t).
Consider the system of two coupled differential equations: y-cx + dy, x-ax + by, with the equilibrium solution (xe,ye) = (0,0) (a) Rewrite the coupled system as a matrix differential equation and identify the matrix A. Obtain a general solution to the matrix differential equation in terms of eigenvectors and eigenvalues of A. Justify your answer (b) Classify possible types and stability of the equilibrium with dependence on the eigenvalues of A. (Note: You are not asked to compute the...
Express the system of differential equations in matrix notation x – 4x + y - (cos t)x = 0 y"+y" - t?x' + 3y'+e-2x = 0 Which of the following sets of definitions allows the given system to be written as an equivalent system in normal form using only the new variables? OA. Xi =X, X2 = X". X3 = y, Xa =y" O B. *= x, X2 = x', *3 = y, X4 =y', X5 =y" OC. *1 =...
Solve the following differential equation using
variation of parameters.
d yt) 2 dy() +7- + 10y() u(t) dt dt2 y(0) 0, y'(0) = 3
d yt) 2 dy() +7- + 10y() u(t) dt dt2 y(0) 0, y'(0) = 3
Along with x1' please solve for
x2'. Thanks!
Transform the given differential equation into an equivalent system of first-order differential equations. y' (t) + 5y' (t) - 6ty(t) = 6 cost Let x, = y and X, Ey. Complete the differential equation for X.
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