Final answer:
To find the second solution y2(x) and a particular solution yp(x) using the reduction of order method, start with the known first solution y1(x) and follow the steps provided. The general solution can then be obtained by substituting the particular solution and the first solution into the differential equation.
Step-by-step explanation:
To find the second solution y2(x) of the homogeneous equation and a particular solution yp(x) of the given nonhomogeneous equation using the method of reduction of order, we start with the known first solution y1(x) = e^(-4x).
Step 1: Assume the second solution in the form y2(x) = u(x) * y1(x), where u(x) is a new function.
Step 2: Substitute y2(x) = u(x) * y1(x) in the differential equation and simplify to obtain a second-order linear homogeneous equation.
Step 3: Solve the resulting equation for u(x) using any appropriate method, such as integrating factor, variation of parameters, or reduction of order.
Step 4: Once u(x) is found, the second solution y2(x) can be obtained by multiplying u(x) with the first solution y1(x).
Step 5: To determine the particular solution yp(x) of the given nonhomogeneous equation, substitute y(x) = yp(x) + c1 * y1(x) into the differential equation, where c1 is an arbitrary constant.
Step 6: Simplify and solve for yp(x).
Step 7: The general solution is y(x) = yp(x) + c1 * y1(x), where c1 is the constant from step 5.