# Step-by-Step Solution: Difference Equations Let's analyze and solve the given difference equations. --- ## Problem 3 **Equation:** \[ y(n) - 0.1y(n-2) = x(n) \] ### Step 1: Homogeneous Solution Set \(x(n) = 0\): \[ y_h(n) - 0.1y_h(n-2) = 0 \] Assume \(y_h(n) = r^n\): \[ r^n - 0.1r^{n-2} = 0 \implies r^n = 0.1r^{n-2} \] \[ r^2 = 0.1 \implies r = \pm \sqrt{0.1} \] So, the homogeneous solution is: \[ y_h(n) = C_1 \left(\sqrt{0.1}\right)^n + C_2 \left(-\sqrt{0.1}\right)^n \] ### Step 2: Particular Solution Assume a particular solution of the form \(y_p(n) = K x(n)\): Plug into the original equation: \[ K x(n) - 0.1 K x(n-2) = x(n) \] \[ K x(n) - 0.1 K x(n-2) = x(n) \] Match coefficients: \[ K = 1, \quad -0.1K = 0 \implies K = 1 \] But this only works if \(x(n-2) = 0\), i.e., for an impulse input. For a general \(x(n)\), use the **Z-transform** (optional for tutor mode). In general, the particular solution depends on the form of \(x(n)\). ### Step 3: General Solution \[ y(n) = C_1 \left(\sqrt{0.1}\right)^n + C_2 \left(-\sqrt{0.1}\right)^n + y_p(n) \] --- ## Problem 5 **Equation:** \[ y(n) + 3y(n-1) = x(n) + x(n-1) \] ### Step 1: Homogeneous Solution Set \(x(n) = 0\): \[ y_h(n) + 3y_h(n-1) = 0 \] Assume \(y_h(n) = r^n\): \[ r^n + 3r^{n-1} = 0 \implies r + 3 = 0 \implies r = -3 \] So, \[ y_h(n) = C(-3)^n \] ### Step 2: Particular Solution Assume \(x(n)\) is a step input, i.e., \(x(n) = 1\) for \(n \geq 0\): Try \(y_p(n) = A\): \[ A + 3A = 1 + 1 \implies 4A = 2 \implies A = 0.5 \] So, for a step input, \[ y_p(n) = 0.5 \] ### Step 3: General Solution \[ y(n) = C(-3)^n + y_p(n) \] If \(x(n)\) is more general, use convolution or Z-transform. --- ## Summary Table | Eq. | Homogeneous Solution | Particular Solution (for step input) | General Solution | |-----|--------------------------------------|--------------------------------------|---------------------------------------| | 3 | \(C_1 (\sqrt{0.1})^n + C_2 (-\sqrt{0.1})^n\) | Depends on \(x(n)\) | Homogeneous + Particular | | 5 | \(C(-3)^n\) | \(0.5\) | \(C(-3)^n + 0.5\) | --- ## Next Steps - If you have specific initial conditions or \(x(n)\), plug those in to solve for constants. - For other forms of \(x(n)\), specify and we can find \(y_p(n)\). --- **Let me know if you want to consider a specific input or initial conditions.**