Understanding the Differential Equation
The given differential equation is (D - 2)^2 y = 8e^(2x), where D represents the differentiation operator with respect to x.
Step 1: Find the Complementary Solution (yc)
- Solve the homogeneous equation: (D - 2)^2 y = 0.
- The characteristic equation is (r - 2)^2 = 0.
- This results in a repeated root r = 2.
- The complementary solution is: yc = C1 e^(2x) + C2 x e^(2x).
Step 2: Find the Particular Solution (yp)
- For the non-homogeneous part 8e^(2x), use the method of undetermined coefficients.
- Since e^(2x) is part of the complementary solution, we need to multiply by x^2 to find yp.
- Assume yp = Ax^2 e^(2x).
- Differentiate yp:
- D(yp) = 2Ax e^(2x) + Ax^2(2e^(2x)) = (2Ax + 2Ax^2)e^(2x).
- D^2(yp) = 2A e^(2x) + 4Ax e^(2x) + 2Ax(2e^(2x)) = (2A + 6Ax)e^(2x).
- Substitute yp into the left side of the original equation to find A:
- (D - 2)^2(yp) = 8e^(2x).
- Solve for A, yielding A = 1.
Step 3: Combine Solutions
- The particular solution is yp = x^2 e^(2x).
- The general solution is: y = yc + yp = C1 e^(2x) + C2 x e^(2x) + x^2 e^(2x).
Conclusion
The solution to the differential equation (D - 2)^2 y = 8e^(2x) is:
y = C1 e^(2x) + C2 x e^(2x) + x^2 e^(2x), where C1 and C2 are constants determined by initial conditions.