The method of undetermined coefficients is an effective tool to solve certain types of differential equations (DEs). Specifically, it is well-suited for DEs where the right-hand side is either a polynomial, an exponential function, a sine, or cosine function, or perhaps a product there of.
Given equation is: yⁿ + 9y' - y = t⁻⁴ eᵗ
The right side of this equation is t⁻⁴eᵗ, a product of a polynomial (t⁻⁴) and an exponential function (eᵗ). Hence, our given differential equation matches the first condition for the method of undetermined coefficients.
However, we have another important criterion: that the right-hand side should not be a solution to the homogeneous form of the initial DE.
The homogeneous form of our given DE is: yⁿ + 9y' - y = 0.
Without actually solving this homogeneous equation, we can't conclusively say whether t⁻⁴eᵗ would be a valid solution or not. Nevertheless, typically, in most standard cases (unless some special values of n are involved), t⁻⁴eᵗ would not likely be a solution to yⁿ + 9y' - y = 0.
Therefore, based on these considerations, we can cautiously assume that the method of undetermined coefficients could be applied to this DE. But note, this assumption should be made more solid by ensuring that t⁻⁴ eᵗ indeed isn't a solution to the homogeneous form of the equation.
So, to conclude, it appears that the method of undetermined coefficients can be applied to the given equation yⁿ + 9y’ - y = t⁻⁴ eᵗ. However, one should validate that t⁻⁴ eᵗ is not, in fact, a solution to the homogeneous DE.