To solve the second-order differential equation with initial conditions and , we first find the characteristic equation: . The roots of this equation are . This gives us a general solution of the form .

Now, applying the initial conditions, we get . For the first derivative, , and evaluating at leads to . Substituting gives .

Thus, the specific solution is:

This solution will describe an oscillating behavior that decays over time due to the presence of the exponential factor.

For a little fun, if you imagine the graph of this function, it’s almost like a roller coaster that’s slowly sinking into the ground—the peaks and valleys of the oscillations will be more and more subdued as time goes on, “fading” like a distant memory!

If you’re curious about practical uses, such differential equations pop up in a variety of fields like engineering, where they can model damped oscillations such as in car suspensions or electrical circuits. Understanding how these systems behave over time helps engineers design better products and structures that handle energy dissipation efficiently.