(1 point) Consider the initial value problem my′′+cy′+ky=F(t), y(0)=0, y′(0)=0 modeling the motion of a spring-mass-dashpot...

(1 point) Consider the initial value problem my′′+cy′+ky=F(t), y(0)=0, y′(0)=0 modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m=2 kilograms, c=8 kilograms per second, k=80 Newtons per meter, and F(t)=80cos(8t) Newtons. Solve the initial value problem. y(t)= Determine the long-term behavior of the system. Is limt→∞y(t)=0? If it is, enter zero. If not, enter a function that approximates y(t) for very large positive values of t. For very large positive values of t, y(t)≈

Expert Solution

Colby Messinger answered 7 months ago

Consider the initial value problem mx′′+cx′+kx=F(t), x(0)=0, x′(0)=0 modeling the motion of a spring-mass-dashpot system initially at rest...

Consider the initial value problem mx′′+cx′+kx=F(t), x(0)=0, x′(0)=0 modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m=2 kilograms, c=8 kilograms per second, k=80 Newtons per meter, and F(t)=80cos(8t) Newtons. Solve the initial value problem. x(t)= Determine the long-term behavior of the system. Is limt→∞x(t)=0? If it is, enter zero. If not, enter a function that approximates x(t) for very large positive values...

A mass-spring-dashpot system is described by my′′ + cy′ + ky = Fo cos ωt, see...

A mass-spring-dashpot system is described by my′′ + cy′ + ky = Fo cos ωt, see §3.6 Eq. (17). This second-order differential equation has been used in simulations, such as this one at the PhET site: https://phet.colorado.edu/en/simulation/legacy/resonance. For m = 2.53kg, c = 0.502N/(m/s), k = 97.2N/m, Fo = 97.2×0.5N = 48.6N,and ω = 2.6, the equation becomes 2.53y′′ + 0.502y′ + 97.2y = 48.6 cos(ωt) ....................... (1) (a) Given the initial value y(0) = 2.20, y′(0) = 0, solve...

Consider the initial value problem mx′′+cx′+kx=F(t), x(0)=0, x′(0)=0 modeling the motion of a damped mass-spring system initially at...

Consider the initial value problem mx′′+cx′+kx=F(t), x(0)=0, x′(0)=0 modeling the motion of a damped mass-spring system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m=2 kilograms, c=8 kilograms per second, k=80 Newtons per meter, and F(t)=100cos(8t) Newtons. Solve the initial value problem. x(t)= Determine the long-term behavior of the system (steady periodic solution). Is limt→∞x(t)=0? If it is, enter zero. If not, enter a function that approximates x(t) for...

Solve the initial value problem: Y''-4y'+4y=f(t) y(0)=-2, y'(0)=1 where f(t) { t if 0<=t<3 , t+2...

Solve the initial value problem: Y''-4y'+4y=f(t) y(0)=-2, y'(0)=1 where f(t) { t if 0<=t<3 , t+2 if t>=3 }

The motion of a mass-spring system with damping is governed by y''(t) + by'(t) + 64...

The motion of a mass-spring system with damping is governed by y''(t) + by'(t) + 64 y(t) = 0; y(0) = 3, and y'(0) = 0. Find the equation of motion for b = 0,14,16, and 20. .

solve the following initial value problem y''+4y'=g(t),y(0)=0,y' (0)=1 if g(t) is the function which is 1...

solve the following initial value problem y''+4y'=g(t),y(0)=0,y' (0)=1 if g(t) is the function which is 1 on [0,1) and zero elsewhere

1. Consider the initial value problem y′ =1+y/t, y(1)=3 for1≤t≤2. • Show that y(t) = t...

1. Consider the initial value problem y′ =1+y/t, y(1)=3 for1≤t≤2. • Show that y(t) = t ln t + 3t is the solution to the initial value problem. • Write a program that implements Euler’s method and the 4th order Runke-Kutta method for the above initial value problem. Use your program to solve with h = 0.1 for Euler’s and h = 0.2 for R-K. • Include a printout of your code and a printout of the results at each...

Use Laplace transforms to solve the initial value problem: y''' −y' + t = 0, y(0)...

Use Laplace transforms to solve the initial value problem: y''' −y' + t = 0, y(0) = 0, y'(0) = 0, y''(0) = 0.

Consider the following initial value problem. y''−4y = 0, y(0) = 0, y'(0) = 5 (a)...

Consider the following initial value problem. y''−4y = 0, y(0) = 0, y'(0) = 5 (a) Solve the IVP using the characteristic equation method from chapter 4. (b) Solve the IVP using the Laplace transform method from chapter 7. (Hint: If you don’t have the same ﬁnal answer for each part, you’ve done something wrong.)

Consider the following initial value problem dy/dt = 3 − 2t − 0.5y, y (0) =...

Consider the following initial value problem dy/dt = 3 − 2*t − 0.5*y, y (0) = 1 We would like to find an approximation solution with the step size h = 0.05. What is the approximation of y(0.1)?

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