The particular Integral of $(D^{2}-2D)y=e^{x}sin~x$ is equal to:

Question

Accepted Answer

To solve the problem of finding the particular integral of the differential equation $(D^{2}-2D)y = e^{x}\sin x$, we need to apply the method of undetermined coefficients or the method of variation of parameters. However, given the structure of the differential operator and the non-homogeneous term, we will focus on the method of undetermined coefficients.

### Explanation for Option 1:

1. **Differential Operator Analysis**:
   - The operator $D^2 - 2D$ can be rewritten as $(D)(D-2)$.
   - The roots of the auxiliary equation $m^2 - 2m = 0$ are $m = 0$ and $m = 2$.

2. **Form of the Particular Integral**:
   - The non-homogeneous term is $e^x \sin x$, which suggests that the particular integral should be of the form $e^{ax}(A \sin x + B \cos x)$, where $a = 1$ in this case.

3. **Applying the Operator**:
   - We apply the operator $(D^2 - 2D)$ to the assumed form $e^x(A \sin x + B \cos x)$.
   - After differentiating and substituting back, we equate coefficients to solve for $A$ and $B$.

4. **Solving for Coefficients**:
   - By substituting $y = e^x(A \sin x + B \cos x)$ into the differential equation and equating coefficients, we find that $A = -\frac{1}{2}$ and $B = 0$.

5. **Conclusion**:
   - Thus, the particular integral is $y_p = -\frac{1}{2}e^x \sin x$, which matches Option 1.

### Explanation for Other Options:

- **Option 2: $e^x \cos x$**
  - This form does not satisfy the differential equation when substituted back, as it does not account for the sine component in the non-homogeneous term.

- **Option 3: $-\frac{1}{2}e^x \cos x$**
  - This form incorrectly assumes a cosine component, which is not present in the particular integral derived from the given differential equation.

- **Option 4: $\frac{1}{2}e^x \sin x$**
  - This form has the correct trigonometric function but the wrong coefficient. The calculations show that the coefficient should be $-\frac{1}{2}$, not $\frac{1}{2}$.

In conclusion, Option 1, $-\frac{1}{2}e^x \sin x$, is the correct particular integral for the given differential equation, as it satisfies the equation when substituted back and aligns with the method of undetermined coefficients.