Step 1: Identify the Differential Operator
The given differential equation is:
(d-2)^2 y = 8 sin(2x).
Here, (d-2)^2 represents the operator d^2 - 4d + 4, where d denotes differentiation with respect to x.
Step 2: Find the Complementary Solution (yc)
To find the complementary solution, solve the homogeneous equation:
(d-2)^2 y = 0.
- The characteristic equation is r^2 - 4r + 4 = 0.
- This factors to (r-2)^2 = 0, giving a repeated root r = 2.
- Hence, the complementary solution is:
yc = C1 e^(2x) + C2 x e^(2x).
Step 3: Find the Particular Solution (yp)
Next, we find a particular solution for the non-homogeneous part:
8 sin(2x).
Since the right-hand side is a sine function, we assume a particular solution of the form:
yp = A sin(2x) + B cos(2x).
Now, we differentiate yp twice and substitute into the original equation to find A and B.
- First derivative: yp' = 2A cos(2x) - 2B sin(2x).
- Second derivative: yp'' = -4A sin(2x) - 4B cos(2x).
Substituting yp and its derivatives into the left side of the equation gives:
-4A sin(2x) - 4B cos(2x) - 8A sin(2x) - 8B cos(2x) = 8 sin(2x).
This leads to a system of equations to solve for A and B.
Step 4: Final Solution
After solving for A and B, combine the complementary and particular solutions:
y = yc + yp = C1 e^(2x) + C2 x e^(2x) + yp.
Thus, the general solution of the differential equation is obtained.