Question

The roots and their multiplicities to the characteristic equation of the homogeneous version of some nonhomogeneous linear ODE with constant coefficients are given below, and the function f (x) that makes it nonhomogeneous is provided as well. Find the form of the particular solution. r1 = −1, k1 = 3 r2 = 0, k2 = 2 f (x) = e−x

Answer 1

To find the particular solution of a nonhomogeneous linear ODE, use a function that satisfies the nonhomogeneous term and is linearly independent from the homogeneous solutions.

Step-by-step explanation:

The particular solution of a nonhomogeneous linear ODE with constant coefficients can be found by finding a function that satisfies the nonhomogeneous term and is linearly independent from the homogeneous solutions. In this case, the homogeneous solutions are y_1(x) = e^(-x) and y_2(x) = 1. Since e^(-x) is already included in the nonhomogeneous term, we can use the function f(x) = x * e^(-x) as the particular solution.