Understanding the Differential Equation
The given expression y = (1/(D - 2)^2)x^2 involves the operator D, which typically denotes differentiation with respect to x. To solve this equation, we need to interpret the operator and apply it accordingly.
Step 1: Apply the Operator
- The operator D acts on x^2 to find its derivatives.
- We calculate (D - 2)x^2. The operator D will differentiate x^2, and we subtract 2 times x^2.
Step 2: Calculate the First Derivative
- D(x^2) = 2x.
- Therefore, (D - 2)x^2 = 2x - 2x^2.
Step 3: Applying the Operator Again
- Now, we need to apply (D - 2) again: (D - 2)(2x - 2x^2).
- D(2x) = 2, and D(-2x^2) = -4x.
- Thus, (D - 2)(2x - 2x^2) = 2 - 4x + 4x^2.
Step 4: Final Application of the Operator
- We apply (D - 2) once more to find (D - 2)(2 - 4x + 4x^2).
- D(2) = 0, D(-4x) = -4, and D(4x^2) = 8x.
- Therefore, (D - 2)(2 - 4x + 4x^2) = 0 - 4 + 8x - 8x^2 = -4 + 8x - 8x^2.
Step 5: Formulating the General Solution
- The general solution can be expressed as y = 1/(D - 2)^2 * x^2.
- Since we simplified the differential operation, we can conclude this as our final output.
Conclusion
- The solution involves manipulating the differential operator effectively.
- This process showcases the application of derivatives and the significance of operator algebra in differential equations.