Final answer:
To determine a suitable form of Y(t) for the given differential equation using the method of undetermined coefficients, assume the form of Y(t) as (At^2 + Bt + C)e^t sint and solve for the coefficients by substituting Y(t) into the equation.
Step-by-step explanation:
The given differential equation is y⁴+2 y'''+2 y''=3e⁴ᵗ + 6te⁻⁵ᵗ + eᵗsint. To determine a suitable form for Y(t) if the method of undetermined coefficients is to be used, we need to consider the form of the right-hand side. The right-hand side consists of exponential and trigonometric functions.
Since the right-hand side contains eᵗsint, we will assume the form of Y(t) to be a polynomial multiplied by eᵗsint. Therefore, the form of Y(t) would be Y(t) = (At^2 + Bt + C)e^t sint.
To determine the coefficients A, B, and C, we need to substitute Y(t) into the original differential equation and solve for the coefficients.