Mass-Spring System: General Solution and Value at Pi

The problem describes a mass-spring-damper system with an external force. We need to find the general solution for the displacement and the value of .

First, let’s identify the given parameters and formulate the differential equation.
The general form of the differential equation for a mass-spring-damper system with an external force is:

where:

From the problem statement:

Substitute these values into the differential equation:

This is a second-order linear non-homogeneous differential equation with constant coefficients.
The general solution will be the sum of the complementary solution and the particular solution .

Step 1: Find the complementary solution
The complementary solution is found by solving the homogeneous equation:

The characteristic equation is:

Use the quadratic formula to find the roots:

Since the roots are complex conjugates of the form , the complementary solution is:

Here, and .

Step 2: Find the particular solution
Since the non-homogeneous term is a constant , we can assume a particular solution of the form , where is a constant.
Then, and .
Substitute these into the original differential equation:

So, the particular solution is:

Step 3: Formulate the general solution
The general solution is the sum of the complementary and particular solutions:

Step 4: Use initial conditions to find and
The problem states the initial conditions are and .
First, apply :

Next, find the derivative of :

Using the product rule for the first term:

Now, apply :

Substitute :

Step 5: Write the specific solution
Substitute the values of and back into the general solution:

This is the general solution for given the initial conditions. The question asks for “the general solution”, which usually means with arbitrary constants, but then asks for which implies using the initial conditions. Given the context, it’s likely asking for the specific solution that satisfies the initial conditions.

Step 6: Calculate
Substitute into the specific solution:

We know that and .

The final answer is .