Question

A solution to the difference equation \(u_{x+3} - 3u_{x+1} - 2u_x = 0\) is of the form:

• A. \(u_x = c_1 2^x - (c_2 + c_3 x)(-1)^x\)
• B. \(u_x = (c_1 - c_2)2^x - (c_3 x)(-1)^x\)
• C. \(u_x = c_1 (1 - (-1)^x)2^x + c_2 (-1)^x\)
• D. \(u_x = c_1 2^x - (c_2 + c_3 x)(-1)^{x+1}\)

Hint:

Exam Tip:
For linear difference equations with constant coefficients:

• Assume \(u_x = r^x\).
• Solve the characteristic equation.
• For repeated roots, multiply by \(x\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We have a linear homogeneous difference equation with constant coefficients. We need to find the general solution.

Step 2: Key Formula or Approach:
The characteristic equation is obtained by assuming \(u_x = r^x\).
Substitute into the difference equation: \[ r^{x+3} - 3r^{x+1} - 2r^x = 0 \Rightarrow r^x (r^3 - 3r - 2) = 0 \] So, \(r^3 - 3r - 2 = 0\).

Step 3: Detailed Explanation:
Solve \(r^3 - 3r - 2 = 0\).
Try \(r = 2\): \(8 - 6 - 2 = 0\), so \(r = 2\) is a root.
Factor: \((r - 2)(r^2 + 2r + 1) = (r - 2)(r + 1)^2 = 0\).
So, roots are \(r = 2\) (simple) and \(r = -1\) (double).
The general solution for a double root \(r = -1\) is \((c_2 + c_3 x)(-1)^x\).
For the simple root \(r = 2\), it's \(c_1 2^x\).
Thus, the general solution is: \[ u_x = c_1 2^x + (c_2 + c_3 x)(-1)^x \] Wait, the options have a minus sign before the second term.
Option (A) is \(u_x = c_1 2^x - (c_2 + c_3 x)(-1)^x\).
Since \(c_2\) and \(c_3\) are arbitrary constants, the minus sign can be absorbed.
So, option (A) is the correct form.

Step 4: Final Answer:
Therefore, option (A) is correct.