Final answer:
To solve the differential equation y'' - 8y' + 20y = 200x^2 - 39xe^x, first solve the homogeneous equation for y_h. Then assume forms for y_p that match the terms on the right side of the equation, substitute them in, and solve for coefficients. The general solution is the sum of y_h and y_p.
Step-by-step explanation:
To solve the given differential equation y'' - 8y' + 20y = 200x^2 - 39xe^x using the method of undetermined coefficients, we start by finding the homogeneous solution of the differential equation.
For the homogeneous equation y'' - 8y' + 20y = 0, we solve the characteristic equation r^2 - 8r + 20 = 0, which has complex roots. The general solution of the homogeneous equation is y_h = e^{4x}(Acos(4x) + Bsin(4x)).
The particular solution y_p to the nonhomogenous equation can be assumed as a polynomial function for the 200x^2 term and an exponential function times a polynomial for the -39xe^x term, i.e., yx = Ax^2 + Bx + C and ye = x(Dx + E)e^x. Substituting these into the differential equation and solving for the coefficients gives us the particular solution.
Finally, the general solution is the sum of the homogeneous and particular solutions y = y_h + y_p.