What is the differential-difference equation? b) Solve for 𝑝𝑥 for the following difference equation in probability, 𝑝𝑥+1 − 2𝑝𝑥 +
𝑝𝑥−1 = 0, true for x = 1, 2, . . . ,N − 1, with boundary conditions 𝑝0 = 1, 𝑝𝑁 = 0.

### Differential-Difference Equation

A differential-difference equation is an equation that involves both differential and difference terms. These equations often arise in the modeling of various physical phenomena where there is a dependence on both continuous changes and discrete shifts. They can be of various forms, involving time or spatial shifts and derivatives. A common form of such an equation can be written as:
\[ \frac{d}{dt} f(t) = a f(t) + b f(t - \tau), \]
where \( a \) and \( b \) are constants, and \( \tau \) is a time delay.

### Solving the Difference Equation

We are given the difference equation:
\[ p_{x+1} - 2p_x + p_{x-1} = 0, \]
which is valid for \( x = 1, 2, \ldots, N-1 \), along with the boundary conditions \( p_0 = 1 \) and \( p_N = 0 \).

To solve this, we recognize that the equation is a second-order linear homogeneous difference equation with constant coefficients. The characteristic equation for this difference equation is:
\[ r^2 - 2r + 1 = 0. \]

Solving the characteristic equation, we find:
\[ r^2 - 2r + 1 = (r - 1)^2 = 0 \implies r = 1 \]

Since the characteristic equation has a repeated root \( r = 1 \), the general solution to the difference equation is:
\[ p_x = A + Bx, \]
where \( A \) and \( B \) are constants to be determined by the boundary conditions.

Applying the boundary conditions:
1. \( p_0 = 1 \):
   \[ A + B \cdot 0 = 1 \implies A = 1 \]
   
2. \( p_N = 0 \):
   \[ 1 + BN = 0 \implies BN = -1 \implies B = -\frac{1}{N} \]

Substituting \( A \) and \( B \) back into the general solution, we get:
\[ p_x = 1 - \frac{x}{N}. \]

Thus, the solution to the difference equation \( p_{x+1} - 2p_x + p_{x-1} = 0 \) with the given boundary conditions is:
\[ p_x = 1 - \frac{x}{N}. \]

### Summary

The difference equation \( p_{x+1} - 2p_x + p_{x-1} = 0 \) with boundary conditions \( p_0 = 1 \) and \( p_N = 0 \) has the solution:
\[ p_x = 1 - \frac{x}{N}. \]

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Would you like details or have any questions on this topic? Here are some related questions that you might find interesting:

1. What are the main types of differential-difference equations?
2. How are differential-difference equations applied in real-world scenarios?
3. Can you explain the method of solving higher-order difference equations?
4. How do boundary conditions affect the solutions of difference equations?
5. What are some common methods for solving non-homogeneous difference equations?
6. What is the role of characteristic equations in solving difference equations?
7. How do we handle difference equations with variable coefficients?
8. Can we use numerical methods to solve difference equations?

**Tip:** When solving difference equations, always check the characteristic equation first to determine the form of the general solution.

Learn how to solve the difference equation p_{x+1} - 2p_x + p_{x-1} = 0 with boundary conditions p_0 = 1 and p_N = 0. This problem involves second-order linear homogeneous difference equations and characteristic equation methods. Suitable for undergraduate students studying difference equations.

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