Solve the nonhomogeneous differential equation y'' - y = 6.

Step 1: Solve the associated homogeneous equation.

First consider:
y'' - y = 0

The characteristic equation is:
r^2 - 1 = 0

Factor:
(r - 1)(r + 1) = 0

So the roots are:
r = 1 and r = -1

Therefore, the complementary solution is:
y_h = C1e^x + C2e^(-x)

Step 2: Find a particular solution.

The right side of the original equation is the constant 6.

Try a constant particular solution:
y_p = A

Step 3: Differentiate the proposed particular solution.

If y_p = A, then:
y_p' = 0
y_p'' = 0

Step 4: Substitute into the differential equation.

The equation is:
y'' - y = 6

Substitute y_p = A and y_p'' = 0:

0 - A = 6

Therefore:
A = -6

So a particular solution is:
y_p = -6

Step 5: Combine the homogeneous and particular solutions.

The general solution to the nonhomogeneous equation is:
y = y_h + y_p

Therefore:
y = C1e^x + C2e^(-x) - 6

Step 6: Check the solution.

Differentiate twice:
y'' = C1e^x + C2e^(-x)

Then:
y'' - y = [C1e^x + C2e^(-x)] - [C1e^x + C2e^(-x) - 6]

The exponential terms cancel:
y'' - y = 6

Therefore, the general solution is y = C1e^x + C2e^(-x) - 6.