91Ó°ÊÓ

To solve a differential equation, we often start by finding the **homogeneous solution**. For the equation \( y'' + y = 0 \), we look for solutions to the related homogeneous equation. This means ignoring any external forces or disturbances, like the Dirac delta function here.

The equation \( y'' + y = 0 \) leads us to a characteristic equation: \( r^2 + 1 = 0 \). Solving this gives the roots \( r = \pm i \). These roots are complex, indicating oscillatory solutions, which naturally occur in many physical systems.

Using these roots, the general form of the homogeneous solution is:

• \( y_h(t) = A \cos(t) + B \sin(t) \)

Here, \( A \) and \( B \) are constants that we'll determine through initial conditions or other boundary requirements.

This oscillatory behavior is typical for systems that have no damping or forcing other than internal spring-like forces.

Next, we focus on the **particular solution** of the differential equation. This solution deals with the equation's non-homogeneous part, specifically the Dirac delta function \( \delta(t - 2\pi) \).

The Dirac delta function models an instantaneous impulse at \( t = 2\pi \). To find the particular solution, we incorporate the Heaviside step function:

• \( y_p(t) = C(t - 2\pi)H(t - 2\pi) \)

This approach ensures the solution activates post-impulse. When we substitute back into the differential equation, we find that \( C = 1 \), making the particular solution:

• \( y_p(t) = (t - 2\pi)H(t - 2\pi) \)

This solution expresses how our system changes immediately after \( t = 2\pi \) due to the impulse force.

Applying **initial conditions** is crucial for determining the constants in our general solution. They're like initial clues provided in a mystery, allowing us to find a unique solution.

For this problem, initial conditions are given as \( y(0) = 0 \) and \( y'(0) = 1 \). We substitute these into our combined solution \( y(t) = y_h(t) + y_p(t) \) to find \( A \) and \( B \).

The complete solution after applying these conditions:

• \( y(t) = A \cos(t) + B \sin(t) + (t - 2\pi)H(t - 2\pi) \)

Substituting initial values:

• At \( t=0 \), \( y(0) = 0 \implies A = 0 \)
• Deriving gives \( y'(0) = 1 \implies B = 1 \)

Thus, the specific solution becomes:

• \( y(t) = \sin(t) + (t - 2\pi)H(t - 2\pi) \)

These conditions ensure our solution fits exactly onto the system's constraints at the start.

The **Dirac delta function** is a key player in many applied math problems, representing an instantaneous impulse or spike at a specific point. For continuous function space, it's like a loud clap amidst silence.

In the differential equation \( y'' + y = \delta(t-2\pi) \), the delta function activates only when \( t = 2\pi \). After that instant, it effectively turns off, but leaves lasting effects, akin to shocking a spring in motion.

The properties of the Dirac delta function include:

• It is zero everywhere except at one point where it is undefined but its integral over its domain is 1.
• Used in systems to model sudden forces or inputs.

Linking it with the Heaviside step function helps mathematically represent the changes post impulse, showcasing its practical significance in physics, engineering, and control systems.