91Ó°ÊÓ

Initial value problems are a type of differential equation problem where the solution is determined by specific conditions given at a starting point. These conditions are known as initial values.
In our exercise, we dealt with a second-order differential equation with two initial conditions:

• The initial velocity: \( s'(0) = 100 \)
• The initial position: \( s(0) = 0 \)

These values define the unique solution to our differential equation by specifying the state of the system at time \( t = 0 \). This is essential because without these conditions, a differential equation may have infinitely many solutions.
By applying the initial conditions, we can solve for the unknown constants which appear during the integration process. This pinpoints the exact path of the system from the multitude of possible solutions.

Trigonometric functions, such as sine and cosine, are fundamental in differential equations, especially when dealing with oscillatory systems. In our problem, we encountered the function \( \sin(2t - \frac{\pi}{2}) \), which illustrates a wave-like behavior.
Trigonometric functions:

• Model periodic phenomena like waves and oscillations
• Switch roles in integration: integrating \( \sin(x) \) gives \( -\cos(x) \) and integrating \( \cos(x) \) gives \( \sin(x) \)

These functions have unique properties, such as phase shifts represented by adding or subtracting angles (e.g., \(-\frac{\pi}{2}\)), which affect their graphs' positions along the time axis.
Such shifts are crucial in accurately representing physical systems in equations, reflecting real-world scenarios like mechanical vibrations or sound waves.

Integration is a key technique used in solving differential equations. In this exercise, you may notice integrating twice, which is typical of second-order differential equations.
During integration:

• Initial integration solved for the velocity function, \( \frac{ds}{dt} = 2 \cos(2t - \frac{\pi}{2}) + C_1 \), where constants arose needing definition through initial conditions.
• Second integration derived the position function. By integrating the velocity, the position \( s(t) \) becomes an expression that includes another constant.

To perform these integrations, we used trigonometric identities and standard integration formulas:

• \( \int \sin(ax) \, dx = -\frac{1}{a} \cos(ax) + C \)
• \( \int \cos(ax) \, dx = \frac{1}{a} \sin(ax) + C \)

Successfully applying these techniques is crucial for finding solutions to differential equations and understanding how different variables interact over time.