Question

Determine the form of a particular

solution for the differential equation.
Do not solve.
y′′+16y′+64y=t²e⁻⁸ᵗ+e⁻⁸ᵗ
The form of a particular solution is
yp​(t)=____
(Do not use d, D, e, E, i, or I as arbitrary constants since these letters already have defined meanings.)

Answer 1

To determine the form of a particular solution for the given differential equation, we can examine the right-hand side of the equation, which consists of two terms: t²e⁻⁸ᵗ and e⁻⁸ᵗ. We need to find a particular solution of the same form as the right-hand side.

Step-by-step explanation:

To determine the form of a particular solution for the given differential equation, we can examine the right-hand side of the equation, which consists of two terms: t²e⁻⁸ᵗ and e⁻⁸ᵗ. We need to find a particular solution of the same form as the right-hand side. Since the terms involve t and an exponential function, we can try a particular solution of the form yp(t) = At²e⁻⁸ᵗ + Be⁻⁸ᵗ. Here, A and B are arbitrary constants that we need to determine.