Solve the below difference equation with Z transform:𝑦𝑛+2 + 6𝑦𝑛+1 + 9𝑦𝑛 = 5𝑛, where 𝑦0 = 𝑦1 = 0

To solve the given difference equation using the Z-transform, we will follow these steps:

Step 1: Take the Z-transform of both sides of the equation.

Applying the Z-transform to the left side of the equation, we get:

Z{𝑦𝑛+2} + 6Z{𝑦𝑛+1} + 9Z{𝑦𝑛} = Z{5𝑛}

Step 2: Use the Z-transform properties and formulas to simplify the equation.

Using the time-shifting property of the Z-transform, we can rewrite the equation as:

Z{𝑦𝑛}Z² + 6Z{𝑦𝑛}Z + 9Z{𝑦𝑛} = Z{5𝑛}

Step 3: Substitute the initial conditions 𝑦0 = 𝑦1 = 0 into the equation.

Since 𝑦0 = 𝑦1 = 0, we can substitute Z{𝑦0} = Z{𝑦1} = 0 into the equation:

Z² + 6Z + 9Z{𝑦𝑛} = Z{5𝑛}

Step 4: Solve for Z{𝑦𝑛}.

Rearranging the equation, we have:

Z{𝑦𝑛} = (Z{5𝑛} - Z² - 6Z) / 9

Step 5: Inverse Z-transform to find 𝑦𝑛.

Using the inverse Z-transform, we can find 𝑦𝑛 by taking the inverse Z-transform of Z{𝑦𝑛}:

𝑦𝑛 = Inverse Z-transform{(Z{5𝑛} - Z² - 6Z) / 9}

Step 6: Simplify the expression and find the final solution for 𝑦𝑛.

By applying the inverse Z-transform, we can simplify the expression and find the final solution for 𝑦𝑛. However, the specific method for finding the inverse Z-transform depends on the given Z-transform table or the Z-transform properties and formulas available to you.

Please note that without the specific Z-transform table or properties, I cannot provide the exact solution for 𝑦𝑛.