Without solving for the undetermined coefficients, the correct form of a particular solution of the differential equation \( y^{\prime \prime}+4 y^{\prime}-2 y=x^{2}+4 \) \( y_{p}=A x+B \) \( y_{p}=c_{1} e^{a x}+c_{2} x e^{b x} \) \( y_{p}=A \) \( y_{p}=c_{1} e^{a x}+c_{2} e^{b x} \) \( y_{p}=A x^{2}+B x+C \)

To determine the correct form of a particular solution \( y_p \) for the differential equation \[ y^{\prime \prime} + 4y^{\prime} - 2y = x^2 + 4, \] we need to analyze the right-hand side of the equation, which is \( x^2 + 4 \). ### Step 1: Identify the form of the non-homogeneous term The right-hand side \( x^2 + 4 \) is a polynomial of degree 2. Therefore, we typically choose a particular solution that is also a polynomial of the same degree. ### Step 2: Choose the form of the particular solution Given that the non-homogeneous term is a polynomial of degree 2, the appropriate form for the particular solution \( y_p \) would be: \[ y_p = Ax^2 + Bx + C, \] where \( A \), \( B \), and \( C \) are constants to be determined. ### Step 3: Evaluate the other options 1. **\( y_p = Ax + B \)**: This is a linear polynomial and would not be sufficient to match the quadratic term \( x^2 \). 2. **\( y_p = c_1 e^{ax} + c_2 x e^{bx} \)**: This form is used for non-homogeneous terms involving exponentials, which does not apply here. 3. **\( y_p = A \)**: This is a constant solution and cannot account for the polynomial terms. 4. **\( y_p = c_1 e^{ax} + c_2 e^{bx} \)**: Similar to the previous exponential forms, this does not match the polynomial nature of the right-hand side. 5. **\( y_p = Ax^2 + Bx + C \)**: This is the correct choice as it matches the degree of the polynomial on the right-hand side. ### Conclusion The correct form of a particular solution for the given differential equation is: \[ y_p = Ax^2 + Bx + C. \]