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[4 pt) The EOM for a harmonically excited undamped system can be written as x'+w+*+/cos (cor)...
Problem 1.Consider the harmonically forced undamped oscillator described by the following ODE:mx′′+kx=F0cosωt, k >0, m >0,...
Problem 1.Consider the harmonically forced
undamped oscillator described by the following ODE:mx′′+kx=F0cosωt,
k >0, m >0, ω >0, F0∈R.
Problem 1. Consider the harmonically forced undamped oscillator described by the following ODE: mx" + kx = Fo cos wt, k > 0, m > 0,w > 0, F0 E R. (1) a) Suppose wa #k/m. Find the general solution of the ODE ). b) Consider the initial value problem of the ODE () with initial conditions x(0) = 0 and...
-(1/4)*cos(2*x)+(1/25)*cos(5*x) incorrect - Cod20) + co(52) At least one of the answers above is NOT correct....
-(1/4)*cos(2*x)+(1/25)*cos(5*x) incorrect - Cod20) + co(52) At least one of the answers above is NOT correct. 2 of the questions remain unanswered. (1 pt) Consider the nonhomogeneous initial-boundary-value problem. Ut – Uzx = cos 2x – cos 5x uz (0,t) = uz (7,t) = 0 u(x,0) = cos 3x The Fourier method suggests the following form for a solution: u(x, t) = LT,(6) n=0 The solution (written in simplest form) is u(x, t) = Describe the steady state temperature distribution:...
Q.3 (Applications of Linear Second Order ODE): Consider the 'equation of motion given by ODE d²x...
Q.3 (Applications of Linear Second Order ODE): Consider the 'equation of motion given by ODE d²x 102 +w²x = Focos(yt) where Fo and wty are constants. Without worrying about those constants, answer the questions (a)-(b). (a) Show that the general solution of the given ODE is [2 pts] Fo x(t) == xc + Ip = ci cos(wt) + C2 sin(wt) +- W2 - 92 cos(7t). (b) Find the values of ci and c2 if the initial conditions are x(0) =...
Previous Problem Problem List Next Problem (1 point) Note WeBWork will interpret acos(z) as cos (z),...
Previous Problem Problem List Next Problem (1 point) Note WeBWork will interpret acos(z) as cos (z), so in order to write a times cos(z) you need to type a cos(z) or put a space between them. The general solution of the homogeneous differential equation can be written as e-acos(x)+bsin(z) where a, b are arbitrary constants and 1r is a particular solution of the nonhomogeneous equation By superposition, the general solution of the equation y" + ly-2ez is บ-Uc + so...
4. a) The one dimensional wave equation for the variable y(z, t) can be written as:...
4. a) The one dimensional wave equation for the variable y(z, t) can be written as: azz czacz where w is the angular frequency (rad/s), k is the wavenumber (rad/m) t is time (s). Show that y(z, t) = 12sin(wt + kz) - 24sin(wt - kz) is a valid solution. (15 marks) b) If a string is fixed at z = Om and at z = 2.4m, and its displacement when vibrating in its fundamental mode is given by: y(z,...
A. 1st Order Systems Consider the spring-damper system shown in Figure 1. Figure 1: spring-damper system...
A. 1st Order Systems Consider the spring-damper system shown in Figure 1. Figure 1: spring-damper system (1)Draw FBD and deduce EOM. Clearly state your assumptions. (2)Cast EOM as an ODE in standard form; write the time constant T and the forcing function f(t) in terms of k,c, f*(1) (3) Write the solution x(t) as the sum x(t) = x (1)+x,() and do the following: a) give the name of x (1) b) write the equation that x(!) must satisfy and...
1.- Starting from the differential equation for a 1-degree of freedom system with mass M, damping...
1.- Starting from the differential equation for a 1-degree of freedom system with mass M, damping c and spring stiffness k: a.- Show that the particular solution for the equation with an applied force fo cos(ot), i.e., Mä+ci+kx=f, cos(or) can be expressed as x )= A cos(ot) + A, sin(or) and find the values of A, and A, that solve the differential equation in terms of M, c, k and fo. 5 points. b. Use the result from part a...
5. The solution to the system X-Ax can be written as etAX(O), where the matrix function etA İS de...
5. The solution to the system X-Ax can be written as etAX(O), where the matrix function etA İS defined by the series etAA)tA) Given the matrix tA- etA (part of Lebl 3.8.2)? 3t t t 3t , what is 3! 2!
5. The solution to the system X-Ax can be written as etAX(O), where the matrix function etA İS defined by the series etAA)tA) Given the matrix tA- etA (part of Lebl 3.8.2)? 3t t t 3t , what is...
Mark each statement True or False. Justify each answer. a. A homogeneous system of equations can...
Mark each statement True or False. Justify each answer. a. A homogeneous system of equations can be inconsistent. Choose the correct answer below. O A. True. A homogeneous equation can be written in the form Ax o, where A is an mxn matrix and 0 is the zero vector in R". Such a system Ax -0 always has at least one solution, namely x-0. Thus, a homogeneous system of O B. True. A homogeneous equation cannot be written in the...
(1 point) Consider the equation for the charge on a capacitor in an LRC circuit which is linear w...
(1 point) Consider the equation for the charge on a capacitor in an LRC circuit which is linear with constant coefficients d2 dt2 First we will work on solving the corresponding homogeneous equation. Divide through the equation by the coefficient on and find the auxiliary equation (using m as your variable) mA2+16m+63 - 0 which has roots The solutions of the homogeneous equation are e^(-7t),engr d2 dt2 di dt Now we are ready to solve the nonhomogeneous equation 4 +...
Problem 1.Consider the harmonically forced
undamped oscillator described by the following ODE:mx′′+kx=F0cosωt,
k >0, m >0, ω >0, F0∈R.
Problem 1. Consider the harmonically forced undamped oscillator described by the following ODE: mx" + kx = Fo cos wt, k > 0, m > 0,w > 0, F0 E R. (1) a) Suppose wa #k/m. Find the general solution of the ODE ). b) Consider the initial value problem of the ODE () with initial conditions x(0) = 0 and...
-(1/4)*cos(2*x)+(1/25)*cos(5*x) incorrect - Cod20) + co(52) At least one of the answers above is NOT correct. 2 of the questions remain unanswered. (1 pt) Consider the nonhomogeneous initial-boundary-value problem. Ut – Uzx = cos 2x – cos 5x uz (0,t) = uz (7,t) = 0 u(x,0) = cos 3x The Fourier method suggests the following form for a solution: u(x, t) = LT,(6) n=0 The solution (written in simplest form) is u(x, t) = Describe the steady state temperature distribution:...
Q.3 (Applications of Linear Second Order ODE): Consider the 'equation of motion given by ODE d²x 102 +w²x = Focos(yt) where Fo and wty are constants. Without worrying about those constants, answer the questions (a)-(b). (a) Show that the general solution of the given ODE is [2 pts] Fo x(t) == xc + Ip = ci cos(wt) + C2 sin(wt) +- W2 - 92 cos(7t). (b) Find the values of ci and c2 if the initial conditions are x(0) =...
Previous Problem Problem List Next Problem (1 point) Note WeBWork will interpret acos(z) as cos (z), so in order to write a times cos(z) you need to type a cos(z) or put a space between them. The general solution of the homogeneous differential equation can be written as e-acos(x)+bsin(z) where a, b are arbitrary constants and 1r is a particular solution of the nonhomogeneous equation By superposition, the general solution of the equation y" + ly-2ez is บ-Uc + so...
4. a) The one dimensional wave equation for the variable y(z, t) can be written as: azz czacz where w is the angular frequency (rad/s), k is the wavenumber (rad/m) t is time (s). Show that y(z, t) = 12sin(wt + kz) - 24sin(wt - kz) is a valid solution. (15 marks) b) If a string is fixed at z = Om and at z = 2.4m, and its displacement when vibrating in its fundamental mode is given by: y(z,...
A. 1st Order Systems Consider the spring-damper system shown in Figure 1. Figure 1: spring-damper system (1)Draw FBD and deduce EOM. Clearly state your assumptions. (2)Cast EOM as an ODE in standard form; write the time constant T and the forcing function f(t) in terms of k,c, f*(1) (3) Write the solution x(t) as the sum x(t) = x (1)+x,() and do the following: a) give the name of x (1) b) write the equation that x(!) must satisfy and...
1.- Starting from the differential equation for a 1-degree of freedom system with mass M, damping c and spring stiffness k: a.- Show that the particular solution for the equation with an applied force fo cos(ot), i.e., Mä+ci+kx=f, cos(or) can be expressed as x )= A cos(ot) + A, sin(or) and find the values of A, and A, that solve the differential equation in terms of M, c, k and fo. 5 points. b. Use the result from part a...
5. The solution to the system X-Ax can be written as etAX(O), where the matrix function etA İS defined by the series etAA)tA) Given the matrix tA- etA (part of Lebl 3.8.2)? 3t t t 3t , what is 3! 2!
5. The solution to the system X-Ax can be written as etAX(O), where the matrix function etA İS defined by the series etAA)tA) Given the matrix tA- etA (part of Lebl 3.8.2)? 3t t t 3t , what is...
Mark each statement True or False. Justify each answer. a. A homogeneous system of equations can be inconsistent. Choose the correct answer below. O A. True. A homogeneous equation can be written in the form Ax o, where A is an mxn matrix and 0 is the zero vector in R". Such a system Ax -0 always has at least one solution, namely x-0. Thus, a homogeneous system of O B. True. A homogeneous equation cannot be written in the...
(1 point) Consider the equation for the charge on a capacitor in an LRC circuit which is linear with constant coefficients d2 dt2 First we will work on solving the corresponding homogeneous equation. Divide through the equation by the coefficient on and find the auxiliary equation (using m as your variable) mA2+16m+63 - 0 which has roots The solutions of the homogeneous equation are e^(-7t),engr d2 dt2 di dt Now we are ready to solve the nonhomogeneous equation 4 +...
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