Question

1.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 =2 f(x)=x³ ​

2. 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

Answer 1

To find the particular solution for a nonhomogeneous linear ODE with constant coefficients, use the method of undetermined coefficients. Assume a specific form of the particular solution that matches the form of the nonhomogeneous term and solve for the unknown coefficients.

Step-by-step explanation:

To find the particular solution for a nonhomogeneous linear ODE with constant coefficients, we can use the method of undetermined coefficients. We look for a form of the particular solution that matches the form of the nonhomogeneous term. For the first problem, since the nonhomogeneous term is a cubic polynomial, we can assume the particular solution to be of the form r
`Ax^3 + Bx^2` the second problem, since the nonhomogeneous term is a quadratic polynomial, we can assume the particular solution to be of the form A
`x^2`Bx + C. We substitute these forms into the ODE and solve for the unknown coefficients.