Differential Equations

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  • Q* Consider the initial value problem: y``` + 2y`` - 5y' -6y = 0, y(0) = 2, y'(0) = 6, y``(0) = 0
  • A* We find the characteristics equation r^3 + r^2 - 5r - 6 = (r + 1)(r + 3)(r - 2). Then the solution is y = Ae^{-t} + Be^{-3t} + Ce^{2t}. Plug in initial values to solve for the constant.
  • Q* A crude model of the vertical motion y(t) of a car encountering the speed bump with speed V is given by:

y(t) = 0 for t <= -L/2V my`` + ky = {cos(pi*V*t/L) for -L/2V <= t <= L/2V; 0 for t > L/2V} (The absence of a damping term indicates that the car's shock absorbers are broken.) Note that the equations are dependent on time only; as the speed is given as V , we can write space x in terms of time t: x = V t. Solve this initial value problem; take m = k = 1 and L = pi for convenience. Thus show that the formula for oscillatory motion after the car has traversed the speed bump is y(t) = A*sin(t), where A depends on the speed V . The speed bump starts from x = -L/2 to x = L/2 and takes the shape of cos(pi*x/L)

  • A* We have to solve this problem in two parts, the homogeneous equation where my`` + ky = 0, and a particular solution my`` + ky = cos(pi*Vt/L).

For the homogenous part, we use characteristics equation where mr^2 + k = 0, ie, r = sqrt{k/m}*i, where i = sqrt{-1}. Let's call w = sqrt{k/m}. Then the homogeneous solution to the problem is y = Asin(wt) + B cos(wt). Plug in y(-L/2V) = 0 = y(L/2V) to find the constants A and B.

For the particular part, we have to guess the solution. For a cosine function being the force, we guess yp = Ccos(pi*Vt/L) + Dsin(pi*Vt/L). We want to plug this into the differential equation, so we differentiate to get yp`` = -C*(pi*V/L)^2 cos(pi*Vt/L) - D*(pi*V/L)^2 sin(pi*Vt/L). Plug yp and yp`` into the differential equation to get m*yp`` + k*yp = -Cm*(pi*V/L)^2 cos(pi*Vt/L) - Dk*(pi*V/L)^2 sin(pi*Vt/L) + Cm*cos(pi*Vt/L) + Dk*sin(pi*Vt/L) = cos(pi*Vt/L). By matching the coefficients of the sine and the cosine, we find that D = 0, and that -Cm*(pi*V/L)^2 + Cm = 1. We can solve for C.

The final answer to the question is y + yp.