Diff eqn- IVP

The differential equation that governs the forced oscillation is shown below:

0.2 d²y/dt² + 1.2 dy/dt +2y = r(t) where r(t) is the external force.

Given that r(t) = 5 cos 4t with y(0) = 0 . find the equation of motion of the forced oscillations

  1. 👍 0
  2. 👎 0
  3. 👁 140
  1. Normalize the equation by multiplying by 5:
    0.2 d²y/dt² + 1.2 dy/dt +2y = 5 cos(4t) = r(t)
    to:
    d²y/dt² + 6 dy/dt + 10y = 25cos(4t)

    Find the complementary solution:
    m²+6m+10=0
    m=-3±i
    So the solution to the homogeneous equation is:
    yc=e^(-3t)(C1*cos(t)+C2*sin(t))

    Now find the particular solution by undetermined coefficients:
    Assume the particular solution to be:
    yp=Acos(4t)+Bsin(4t)
    and substitute in y of the the original equation:

    d²yp/dt² + 6 dyp/dt + 10yp = 25cos(4t)

    -16Acos(4t)-16Bsin(4t)
    +6(4Bcos(4t)-4Asin(4t))
    +10Acos(4t)+10Bsin(4t)
    =(-6A+24B)cos(4t)+(-24A-6B)sin(4t)
    Compare coefficients of cos(4t) and sin(4t):
    -24A-6B=0 => B=-4A
    -6A+24B=25 => -102A=25 => A=-25/102
    Therefore
    yp(t)=-(25/102)cos(4t)+(100/102)sin(4t)
    (substitute in homogeneous equation to verify that you get 25cos(4t) )

    The general solution is therefore:
    y=yc+yp=e^(-3t)(C1*cos(t)+C2*sin(t))-(25/102)cos(4t)+(100/102)sin(4t)

    Initial conditions:
    To solve the second order problem completely, you'll need two initial conditions. We are givn y(0)=0 at t=0.
    We need another one (such as y'(0)=5 at t=0).

    Substitute the initial conditions into the general solution above and solve for C1 and C2 to give the final solution of the initial value problem.

    1. 👍 0
    2. 👎 0

Respond to this Question

First Name

Your Response

Similar Questions

  1. Calculus

    Suppose that we use Euler's method to approximate the solution to the differential equation 𝑑𝑦/𝑑𝑥=𝑥^4/𝑦 𝑦(0.1)=1 Let 𝑓(𝑥,𝑦)=𝑥^4/𝑦. We let 𝑥0=0.1 and 𝑦0=1 and pick a step size ℎ=0.2.

  2. Differential Equations

    Suppose a small cannonball weighing 16 pounds is shot vertically upward, with an initial velocity v0 = 300 ft/s. The answer to the question "How high does the cannonball go?" depends on whether we take air resistance into account.

  3. Differential Equations

    The velocity v of a freefalling skydiver is well modeled by the differential equation m*dv/dt=mg-kv^2 where m is the mass of the skydiver, g is the gravitational constant, and k is the drag coefficient determined by the position

  4. calculus

    The differential equation below models the temperature of a 87°C cup of coffee in a 17°C room, where it is known that the coffee cools at a rate of 1°C per minute when its temperature is 67°C. Solve the differential equation

  1. Calculus/Math

    The slope field for a differential equation is shown in the figure. Determine the general solution of this equation. The slope field has positive slopes in quadrants 2 and 4 and negative slopes in quadrants 1 and 4. It looks like

  2. Calculus

    Consider the differential equation dy/dx = x^4(y - 2). Find the particular solution y = f(x) to the given differential equation with the initial condition f(0) = 0. Is this y=e^(x^5/5)+4?

  3. Help with differential eqs problem???? (Calculus)

    Consider the differential equation dy/dt=y-t a) Determine whether the following functions are solutions to the given differential equation. y(t) = t + 1 + 2e^t y(t) = t + 1 y(t) = t + 2 b) When you weigh bananas in a scale at the

  4. Calculus

    Consider the differential equation dy/dx = 2x - y. Let y = f(x) be the particular solution to the differential equation with the initial condition f(2) = 3. Does f have a relative min, relative max, or neither at x = 2? Since

  1. Calculus

    For Questions 1–3, use the differential equation given by dx equals xy/3, y > 0. Complete the table of values x −1 −1 −1 0 0 0 1 1 1 y 1 2 3 1 2 3 1 2 3 dy/dx ? ? ? ? ? ? ? ? ? Find the particular solution y = f(x) to the

  2. Calculus!!

    Consider the differential equation given by dy/dx = xy/2. A. Let y=f(x) be the particular solution to the given differential equation with the initial condition. Based on the slope field, how does the value of f(0.2) compare to

  3. math

    Consider the differential equation dy/dx = -1 + (y^2/ x). Let y = g(x) be the particular solution to the differential equation dy/ dx = -1 + (y^2/ x) with initial condition g(4) = 2. Does g have a relative minimum, a relative

  4. Math (Calc) (Differential Equation Solution)

    Some of the curves corresponding to different values of C in the general solution of the differential equation are shown in the graph. Find the particular solution that passes through the point (0, 2). y(x^2+y) = C 2xy +

You can view more similar questions or ask a new question.