Question

Using the Method of Undetermined Coefficients, determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.) y"-8y'+ 16y=20t⁶e⁴ᵗ

What are the roots of the auxiliary equation associated with the given differential equation?
A. The associated auxiliary equation has the two roots (Use a comma to separate answers as needed.)
B. The associated auxiliary equation has the double root
C. There are no roots to the associated auxiliary equation.

Answer 1

The particular solution for the differential equation is y_p(t) = t⁶e⁴ᵗ.

Step-by-step explanation:

The given differential equation is y"-8y'+ 16y=20t⁶e⁴ᵗ. To find the particular solution using the Method of Undetermined Coefficients, we assume a particular solution of the form:

y_p(t) = At⁶e⁴ᵗ

We substitute the assumed solution into the differential equation and equate coefficients:

(20A)e⁴ᵗ = 20t⁶e⁴ᵗ

By comparing the powers of t and the exponential terms, we find A = 1.

Therefore, the particular solution is y_p(t) = t⁶e⁴ᵗ.