Differential Equations

1. Step 1: Start with the given differential equation:

1. This is a first-order linear differential equation involving both and .
2. Step 2: Rearrange the terms to isolate :

1. Now, we have an equation that describes how changes with respect to , in terms of the function and its derivative.
2. Step 3: Notice that the form of the equation suggests that might be related to in some way. Since as , we expect to simplify as well, possibly to a constant.
3. Step 4: Assume that as , the behavior of the functions becomes simpler. Specifically, since , we hypothesize that may behave similarly to an exponential function that decays as does.
4. Step 5: Assume a solution for in the form:

1. This form satisfies the differential equation and the condition that as , because .
2. Step 6: Substitute this proposed solution into the original differential equation to verify that it satisfies the equation. If both sides of the equation balance, then our assumption is correct.

1. After substituting into this equation, we find that it holds true, confirming the correctness of our solution.
2. Step 7: Therefore, the correct relationship is:

The particle’s position is given by:

Step 1: Velocity of the particle. The velocity is the derivative of with respect to time :

Differentiate :

Since , the velocity becomes:

Step 2: Condition for the particle to come to rest. The particle comes to rest when , i.e., when:

The general solution for is:

From this, we solve for :

Let the particle come to rest at two consecutive points corresponding to and . The values of at these points are:

Step 3: Distance between rest points. At the rest points, the position is:

Substitute and :

Using the property of sine:

Thus:

The distance between the two points is:

Since , the absolute value gives:

Conclusion: The particle comes to rest at two points whose distance is:

Step 1: Start with the given differential equation:

Rearranging it:

This is a linear first-order differential equation.

Step 2: Multiply the equation by the integrating factor , which is .

This simplifies to:

Step 3: Integrate both sides of the equation with respect to :

The right-hand side can be integrated by parts:

Thus, we have:

Step 4: Solve for :

Step 5: Use the initial condition to find :

Solving for :

Step 6: Substitute the value of into the equation for :

Step 7: Finally, take the limit as :

Answer: (D)