Final answer:
The characteristic equation for a differential equation determines the roots of the equation. In this case, the characteristic equation is (r - 2)^2 * (r - 4)^2 = 0. To find a homogeneous differential equation, we use the roots of the characteristic equation as the solutions.
Step-by-step explanation:
The characteristic equation for a differential equation is a polynomial equation that determines the roots (or eigenvalues) of the equation. In this case, the characteristic equation is given as (r - 2)^2 * (r - 4)^2 = 0. To find a differential equation assuming it is homogeneous, we use the roots of the characteristic equation as the solutions.
We have two distinct roots, r = 2 and r = 4, each with multiplicity 2. This means that the differential equation will have terms of the form e^(2t), t*e^(2t), e^(4t), and t*e^(4t). By combining these terms with arbitrary coefficients, we can construct a differential equation that satisfies the given characteristic equation.
For example, one possible homogeneous differential equation would be y'' - 6y' + 16y = 0, where y is the dependent variable and primes denote derivatives concerning t.