what is the particular integral of xe^x??

∫xe^x dx = xe^x - e^x + c, where c is a constant

This one should be part of your repertoire of memorized standard integrals.

To find the particular integral of xe^x, we can use the method of undetermined coefficients, which is applicable for finding the particular solution to nonhomogeneous linear differential equations.

Step 1: Find the complementary function:
The complementary function represents the solution to the associated homogeneous equation, which is obtained by setting the nonhomogeneous term to zero. In this case, the associated homogeneous equation is of the form y'' + y' + y = 0.

The characteristic equation for this homogeneous equation is r^2 + r + 1 = 0. Solving this quadratic equation, we get complex conjugate roots: r = (-1 ± sqrt(3)i)/2.

Hence, the complementary function is of the form y_c = Ae^(-x/2) * cos((sqrt(3)x)/2) + Be^(-x/2) * sin((sqrt(3)x)/2), where A and B are arbitrary constants.

Step 2: Determine the form of the particular integral:
Since the nonhomogeneous term is xe^x, we assume that the particular integral takes the form of a polynomial multiplied by e^x. Since xe^x is already in the form of a polynomial multiplied by e^x, we assume a particular integral of the form y_p = (Ax + B)e^x, where A and B are constants to be determined.

Step 3: Calculate the derivatives and substitute into the original differential equation:
Taking the first and second derivatives of y_p:

y_p' = A(e^x + xe^x) + Be^x
y_p'' = A(2e^x + xe^x) + 2Axe^x + Be^x

Substituting these derivatives into the original nonhomogeneous equation y'' + y' + y = xe^x:

(A(2e^x + xe^x) + 2Axe^x + Be^x) + (A(e^x + xe^x) + Be^x) + (Ax + B)e^x = xe^x

Step 4: Simplify and compare coefficients:
Grouping like terms, we get:

(3A + 2Ax + A) e^x + (2A + B)e^x + Axe^x + Bxe^x = xe^x

Comparing coefficients of like terms on both sides, we obtain:

3A + A = 0 => 4A = 0 => A = 0
2A + B = 0 => B = -2A => B = 0

Since A = B = 0, we find that the particular integral is y_p = 0.

Step 5: Determine the general solution:
The general solution of the nonhomogeneous differential equation is the sum of the complementary function (y_c) and the particular integral (y_p):

y = y_c + y_p = Ae^(-x/2) * cos((sqrt(3)x)/2) + Be^(-x/2) * sin((sqrt(3)x)/2) + 0

Therefore, the particular integral of xe^x is simply zero.

Note: Since the original function xe^x is already a solution to the homogeneous equation, it does not contribute to the particular integral.