A certain mass-spring vibrating between two points and its displacement 𝑦 satisfy the following

Question

governing differential equation 𝑑2𝑦
𝑑𝑥2− 2 𝑑𝑦
𝑑𝑥 + 2𝑦 = 𝑒2𝑥 + sin(2𝑥). Find the displacement of
vibrating spring.

Answer 1

To find the displacement of the vibrating spring, we first need to solve the given differential equation:

d^2y/dx^2 - 2(dy/dx) + 2y = e^2x + sin(2x)

The general solution of this differential equation can be written as:

y(x) = y_h(x) + y_p(x)

where y_h(x) is the homogeneous solution and y_p(x) is the particular solution.

First, let's find the homogeneous solution by setting the right side of the equation to zero:

d^2y_h/dx^2 - 2(dy_h/dx) + 2y_h = 0

The characteristic equation corresponding to this homogeneous differential equation is:

r^2 - 2r + 2 = 0

Solving this quadratic equation, we get complex roots: r = 1 ± i

Therefore, the homogeneous solution is:

y_h(x) = e^x(Acos(x) + Bsin(x))

Next, let's find the particular solution. Since the right side of the equation is e^2x + sin(2x), we try a particular solution of the form:

y_p(x) = C1e^2x + C2sin(2x)

Substitute y_p(x) back into the differential equation and solve for C1 and C2. After solving, we get:

y_p(x) = -e^2x + 0.5sin(2x)

Therefore, the general solution of the differential equation is:

y(x) = e^x(Acos(x) + Bsin(x)) - e^2x + 0.5sin(2x)

where A and B are constants determined by initial conditions of the vibrating spring.

This is the displacement of the vibrating spring.