Find a particular solution $y_{p}$ of the following equation using the Method of Undetermined Coefficients. Primes denote the derivatives with respect to $x$. \[ y^{\prime \prime}-y^{\prime}-2 y=10 x+8 \] A particular solution is $y_{p}(x)=$

Step 1 ：The given differential equation is a second order linear homogeneous differential equation. The method of undetermined coefficients is a way to find a particular solution to such an equation.

Step 2 ：The first step is to find the complementary solution, \(y_c\), of the homogeneous equation (i.e., the equation without the right-hand side). The characteristic equation of the homogeneous equation is \(r^2 - r - 2 = 0\). Solving this quadratic equation will give us the roots, which we can use to write the complementary solution.

Step 3 ：The next step is to guess a form for the particular solution, \(y_p\), based on the right-hand side of the equation. Since the right-hand side is a linear function, we guess that \(y_p\) is also a linear function, i.e., \(y_p = Ax + B\) for some constants \(A\) and \(B\).

Step 4 ：We then substitute \(y_p\) into the original equation and solve for \(A\) and \(B\). This will give us the particular solution.

Step 5 ：\(y_p = A*x + B\)

Step 6 ：\(y_p' = A\)

Step 7 ：\(y_p'' = 0\)

Step 8 ：Substitute \(y_p\), \(y_p'\), and \(y_p''\) into the equation: \(-2*A*x - A - 2*B = 10*x + 8\)

Step 9 ：Solving the equation gives the solution: \(A = -5\), \(B = -\frac{3}{2}\)

Step 10 ：This means that the particular solution to the differential equation is \(y_p = -5x - \frac{3}{2}\)

Step 11 ：Final Answer: The particular solution of the given differential equation is \(y_{p}(x) = \boxed{-5x - \frac{3}{2}}\)