Find the general solution of the following differential equations using the method of undeter-mined coefficientsy′′ − 2y′ + y = ex + x2

The given differential equation is a non-homogeneous second order linear differential equation. The general solution of such an equation is the sum of the general solution of the corresponding homogeneous equation and a particular solution of the non-homogeneous equation.

Step 1: Solve the homogeneous equation The homogeneous equation is y'' - 2y' + y = 0. The characteristic equation is r^2 - 2r + 1 = 0, which can be factored into (r - 1)^2 = 0. Therefore, r = 1 is a root of multiplicity 2. The general solution of the homogeneous equation is y_h = c1e^x + c2xe^x.

Step 2: Find a particular solution of the non-homogeneous equation The non-homogeneous equation is y'' - 2y' + y = e^x + x^2. We guess a particular solution in the form y_p = A*e^x + Bx^2 + Cx + D. Substituting y_p into the non-homogeneous equation and equating coefficients, we can solve for A, B, C, and D.

Step 3: Write down the general solution The general solution of the non-homogeneous equation is the sum of the general solution of the homogeneous equation and the particular solution of the non-homogeneous equation, i.e., y = y_h + y_p.

Note: The actual values of A, B, C, and D depend on the specific form of the non-homogeneous term e^x + x^2. You need to substitute y_p into the non-homogeneous equation and equate coefficients to find A, B, C, and D.