Solve the differential equation using the method of variation of parameters.

Question

Quizplus · Accepted Answer

The method of variation of parameters involves assuming that the solution to the homogeneous equation (found by setting the right-hand side of the differential equation to 0) is of the form y_h(t) = c_1e^{2t} + c_2e^{-t}. Then, we assume that the solution to the non-homogeneous equation is of the form y_p(t) = v_1(t)e^{2t} + v_2(t)e^{-t}, where v_1(t) and v_2(t) are functions to be determined. We substitute y_p(t) into the differential equation and solve for v_1'(t) and v_2'(t). Then, we integrate these expressions to find v_1(t) and v_2(t), and substitute these functions back into the expression for y_p(t). Finally, we add y_h(t) and y_p(t) to get the general solution.

Using this method on the differential equation 9y'' + 14y' + 5y = 3e^{2t}, we find that the general solution is y(t) = c_1e^{t} + c_2e^{5t/3} + (2/9)e^{2t}. The correct answer choice is B.

Solve the Differential Equation Using the Method of Variation of Parameters