Question

In this problem you will use undetermined coefficients to solve the nonhomogeneous equation y′′+2y′+y=6te−ᵗ - 6e−ᵗ−(2t+1) with initial values y(0)=1 and y'(0)=1. Write the characteristic equation for the associated homogeneous equation. (Use r for your variable.)

Answer 1

To find the characteristic equation for the associated homogeneous differential equation y''+2y'+y=0, assume a solution of e^rt, leading to the characteristic equation r^2 + 2r + 1 = 0.

Step-by-step explanation:

The characteristic equation for the associated homogeneous equation of the differential equation y''+2y'+y=0 can be found by assuming a solution of the form ert where r is the variable we are solving for. Substituting y = ert into the homogeneous equation gives us r2ert + 2rert + ert = 0. Dividing through by ert, we obtain the characteristic equation r2 + 2r + 1 = 0.