Final answer:
To solve the nonhomogeneous differential equation, we find the homogeneous solution using the characteristic equation, then assume a particular solution form and determine its constant to find the general solution.
Step-by-step explanation:
To solve the nonhomogeneous differential equation y′′ − 4y′ + 4y = 6e²ᵗ, we need to follow systematic steps involving the characteristic equation and the method of undetermined coefficients or variation of parameters. Firstly, we solve the associated homogeneous equation y′′ − 4y′ + 4y = 0 by finding the roots of its characteristic equation r² − 4r + 4 = 0. The roots are r=2, each with multiplicity 2, giving us the homogeneous solution yh = (C1 + C2t)e²t.
Next, to solve the nonhomogeneous part, we assume a particular solution of the form yp = Ate²t, and substitute back into the differential equation to solve for A, thus obtaining the general solution y = yh + yp.