Particular Integral of the differential equation $(D^{3}+3D^{2}+2D)y=x^{2}$ is:

Question

Accepted Answer

To solve the differential equation $ (D^{3} + 3D^{2} + 2D)y = x^{2} $ and find the particular integral, we need to apply the method of undetermined coefficients or the method of inverse operators. Here, $ D $ represents the differential operator $\frac{d}{dx}$.

### Explanation for Option 2:

1. **Operator Factorization**: The differential operator $ D^{3} + 3D^{2} + 2D $ can be factored as $ D(D+1)(D+2) $. This factorization is crucial because it helps us apply the inverse operator method effectively.

2. **Inverse Operator Method**: To find the particular integral (PI), we apply the inverse of the operator to the right-hand side of the equation, $ x^{2} $. The inverse operator is $ \frac{1}{D(D+1)(D+2)} $.

3. **Applying the Inverse Operator**: 
   - First, apply $ \frac{1}{D} $ to $ x^{2} $, which gives $ \frac{x^{3}}{3} $.
   - Next, apply $ \frac{1}{D+1} $ to $ \frac{x^{3}}{3} $. This involves shifting the function by -1, resulting in $ \frac{x^{3}}{3} - \frac{3x^{2}}{2} $.
   - Finally, apply $ \frac{1}{D+2} $ to the result. This involves another shift by -2, yielding $ \frac{x^{3}}{3} - \frac{3x^{2}}{2} + \frac{7x}{2} $.

4. **Resulting Particular Integral**: After applying all the inverse operators, the particular integral becomes:
   $$
   \frac{1}{2} \left( \frac{x^{3}}{3} - \frac{3x^{2}}{2} + \frac{7x}{2} \right)
   $$
   This matches exactly with Option 2.

### Explanation for Why Other Options are Incorrect:

- **Option 1**: 
  - This option suggests a positive coefficient for the $ x^{2} $ term, which does not align with the shifts applied by the inverse operators. The correct shifts result in a negative coefficient for $ x^{2} $.

- **Option 3**: 
  - This option has a different factor outside the parentheses ($\frac{1}{3}$ instead of $\frac{1}{2}$), which does not match the factor derived from the inverse operator method. Additionally, the coefficients inside the parentheses do not align with the correct shifts.

- **Option 4**: 
  - This option has a negative coefficient for the $ x $ term, which contradicts the positive coefficient derived from the inverse operator method. The shifts applied result in a positive coefficient for $ x $.

In conclusion, Option 2 is the correct answer because it accurately reflects the result of applying the inverse operator method to the differential equation, taking into account the necessary shifts and coefficients.