Can the method of undetermined coefficients together with superposition be applied to find a particular solution of the given equation?

y″−3y′+5y=2cos(8t)−e3tt8+
1

t

Question

Can  the method of undetermined coefficients together with superposition be applied to find a particular solution of the given equation?

y″−3y′+5y=2cos(8t)−e3tt8+
1

t

BlackTom AI · Accepted Answer

To determine whether the method of undetermined coefficients together with superposition can be used, we first restate the problem: we want a particular solution to y'' − 3y' + 5y = 2 cos(8t) − e^{3t} t^8 + 1, and we consider the given option that claims the method cannot be applied because the right-hand side is not the correct type of function.

Let’s recall the standard applicability criteria for the method of undetermined coefficients. This method works well when the nonhomogeneous term (the right-hand side) is composed of simple functions such as polynomials in t, exponentials e^{αt}, sines, and cosines (or finite sums/products thereof, like t^m e^{αt} or t^m cos(βt), t^m sin(βt)), provided α and β do not cause resonance with the characteristic roots of the corresponding homogeneous equation. In particular, the method can handle terms like: P_m(t) e^{αt}, or P_m(t) cos(βt), or P_m(t) sin(βt), where P_m is a polynomial, as long as α ± iβ are not roots of the homogeneous characteristic equation. When a term has a polynomial factor multiplying an exponential or a trigonometric function, we typically adjust the trial solution to have a polynomial of appropriate degree.

Analyzing the given right-hand side components term by term:
- The 2 cos(8t) term is a standard trigonometric forcing. This is of the form P_0(t) cos(βt) with P_0 constant and β = 8, which is compatible with the method, since cos(8t) can be represented as the real part of e^{i8t} and α = 0 does not coincide with any root of the characteristic equation (the characteristic roots are (3 ± i√11)/2, neither purely imaginary nor equal to i8).
- The term − e^{3t} t^8 is a polynomial in t multiplied by an exponential e^{3t}. This fits the form P_m(t) e^{αt} with α = 3 and m = 8. The method of undetermined coefficients can handle such terms by proposing a trial solution of the form e^{3t} times a polynomial of degree 8 (or a polynomial times e^{3t}) as long as α is not a root of the homogeneous equation. Since the homogeneous characteristic roots are (3 ± i√11)/2, α = 3 is not a root, so there is no resonance issue here.
- The remaining constant term (if the ‘+ 1’ at the end is indeed present) is a constant forcing, which is a special case of P_m(t) with degree m = 0 and would be handled by including a constant trial term in the particular solution.

Given these observations, none of the forcing terms violate the typical applicability conditions for the method of undetermined coefficients. Therefore, the assertion that the method cannot be applied because the right-hand side is not the correct type of function is incorrect. The right-hand side contains components (polynomial times exponential, and cosine) that are standard for this method, and superposition allows combining particular solutions corresponding to each component.

In summary, the statement is misleading: the right-hand side is indeed of a type that is compatible with the method of undetermined coefficients (and superposition). The only potential caveat would be if there were resonance (which would require adjustment by multiplying by t), but that is not the case here given the homogeneous roots. Consequently, the reason given in the option does not justify excluding the method.

MAP2302 Lecture Quiz 15

Can the method of undetermined coefficients together with superposition be applied to find a particular solution of the given equation? y″−3y′+5y=2cos(8t)−e3tt8+ 1 t

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