Answer:
To solve the given differential equation using the z-transform, we'll denote the z-transform of a sequence y[n] as Y(z), where z represents the complex variable. The z-transform of the differential equation will allow us to find the expression for Y(z). Here's how we can solve it step-by-step:
Apply the initial conditions:
y[0] = 1 -> Y(z) | z=0 = 1
y[1] = 2 -> Y(z) | z=1 = 2
Shift the equation indices:
y[n + 2] = 4y[n + 1] + 5y[n] -> Y(z) - z^2Y(z) - zY(z) = 4(zY(z) - Y(z)) + 5Y(z)
Simplify and rearrange the equation:
Y(z) - z^2Y(z) - zY(z) = 4zY(z) - 4Y(z) + 5Y(z)
Y(z)(1 - z^2 - z - 4z + 4 + 5) = 0
Y(z)(-z^2 - 5z + 9) = 0
Solve for Y(z):
Y(z) = 0 / (-z^2 - 5z + 9)
= 0, for z ≠ -3 and z ≠ -1
We have a second-order polynomial in the denominator, so let's factor it:
-z^2 - 5z + 9 = -(z - 1)(z + 9)
Therefore, the solutions for Y(z) are:
Y(z) = 0, for z ≠ -3 and z ≠ -1
Find the partial fraction decomposition of Y(z):
Y(z) = A / (z - 1) + B / (z + 9)
To find the values of A and B, let's perform the partial fraction decomposition:
A / (z - 1) + B / (z + 9) = (A(z + 9) + B(z - 1)) / (z - 1)(z + 9)
Equating the numerators, we get:
A(z + 9) + B(z - 1) = 0
Plugging in z = 1, we have:
A(1 + 9) + B(1 - 1) = 2
10A = 2
A = 1/5
Plugging in z = -9, we have:
A(-9 + 9) + B(-9 - 1) = 0
-10B = 0
B = 0
Therefore, the partial fraction decomposition is:
Y(z) = 1/5 / (z - 1)
Apply the inverse z-transform to find y[n]:
Using the z-transform table, we know that the inverse z-transform of 1/5 / (z - 1) is (1/5) * (1^n).
Hence, the solution to the given differential equation is:
y[n] = (1/5) * (1^n) = 1/5, for all values of n.
Therefore, the solution to the given differential equation y[n + 2] = 4y[n + 1] + 5y[n], with initial conditions y[0] = 1 and y[1] = 2, is y[n] =