Question

Consider the following differential equation to be solved by the method of undetermined coefficients. y" + 2y' = 2x + 3 - e⁻²ˣ

Find the complementary function for the differential equation. Yc(x) =
Find the particular solution for the differential equation. Yp(x) =
Find the general solution for the differential equation. y(x) =

Answer 1

The complementary solution for the differential equation is Yc(x) = C1 + C2e^{-2x}. The particular solution Yp(x) involves undetermined coefficients and requires modification due to overlap with Yc(x). The general solution combines Yc(x) and Yp(x).

Step-by-step explanation:

To find the complementary function (Yc(x)) for the differential equation y" + 2y' = 2x + 3 - e-2x, we first consider the homogeneous part of the equation, y" + 2y' = 0. The characteristic equation is r2 + 2r = 0, which factors to r(r + 2) = 0. This gives us roots r = 0 and r = -2. Therefore, Yc(x) = C1 + C2e-2x.

The particular solution (Yp(x)) can be found using the method of undetermined coefficients. For the right side of the equation, we try a particular solution in the form of Yp(x) = Ax + B + Ce-2x. However, since e-2x is already part of the complementary solution, we have to multiply by x to get a new form for the exponential term in the particular solution. Thus, we revise our guess for Yp(x) to be Ax + B + Cxe-2x.

Substituting Yp(x) into the differential equation and solving for A, B, and C, we will get the coefficients that make Yp(x) satisfy the nonhomogeneous differential equation.

The general solution (y(x)) to the differential equation is the sum of the complementary function and the particular solution, y(x) = Yc(x) + Yp(x). This includes the constants from the complementary solution as well as the determined coefficients for the particular solution.

Answer 2

The differential equation y" + 2y' = 2x + 3 - e^{-2x} is solved by finding the complementary function Yc(x), the particular solution Yp(x), and then combining them for the general solution y(x). The complementary function is Yc(x) = C1 + C2e^{-2x}, and the particular solution form should be Yp(x) = Ax + B + Cxe^{-2x}. The general solution is a combination of these two.

Step-by-step explanation:

The given differential equation is y" + 2y' = 2x + 3 - e^{-2x}. To solve this equation, we need to find the complementary function Yc(x), the particular solution Yp(x), and then combine them to get the general solution y(x).

Complementary Function

First, we solve the homogeneous part of the differential equation, which is y" + 2y' = 0. The characteristic equation for this is r2 + 2r = 0, with roots r=0 and r=-2. Therefore, the complementary function is Yc(x) = C1 + C2e^{-2x}.

Particular Solution

To find the particular solution Yp(x), we use the method of undetermined coefficients. Given the right-hand side of the differential equation contains 2x+3 and e^{-2x}, we choose a trial solution in the form Yp(x) = Ax + B + Ce^{-2x}. However, because e^{-2x} is already part of the complementary solution, we need to multiply by x to make it a valid trial solution, hence it becomes Yp(x) = Ax + B + Cxe^{-2x}.

General Solution

The general solution is the sum of the complementary and particular solutions, so it would have the form y(x) = Yc(x) + Yp(x), or y(x) = C1 + C2e^{-2x} + Ax + B + Cxe^{-2x} after determining A, B, and C by substituting Yp(x) into the original differential equation and solving for the coefficients.