Can the method of undetermined coefficients with superposition be used to solve the DE?

y″−y′+y=(7e9t+t)5

a. no, since the coefficients of the DE are not constant

b. no, because the right side of the equation is not the correct type of function

c. no, because the DE is not linear

d. yes

Question

Can the method of undetermined coefficients with superposition be used to solve the DE?

y″−y′+y=(7e9t+t)5

a.  no, since the coefficients of the DE are not constant

b. no, because the right side of the equation is not the correct type of function

c. no, because the DE is not linear

d. yes

BlackTom AI · Accepted Answer

Question restated: Can the method of undetermined coefficients with superposition be used to solve the DE y'' - y' + y = (7 e^{9t} + t)^5?

Answer options:
- a. no, since the coefficients of the DE are not constant
- b. no, because the right side of the equation is not the correct type of function
- c. no, because the DE is not linear
- d. yes

Analysis of each option:
Option a: "no, since the coefficients of the DE are not constant." In fact, the left-hand side is y'' - y' + y, which has constant coefficients. The claim that the coefficients are not constant is incorrect. Since constant-coefficient linear ODEs are precisely the setting where the method of undetermined coefficients is often applicable, this statement mischaracterizes the problem and is false. A student should recognize that constant coefficients do not disqualify the method; they actually enable it.

Option b: "no, because the right side of the equation is not the correct type of function." The right-hand side (7 e^{9t} + t)^5 expands into a finite sum of terms each of which is a polynomial in t multiplied by an exponential e^{k t} for k = 0, 9, 18, 27, 36, 45, arising from the binomial expansion of (7 e^{9t} + t)^5. The standard undetermined-coefficients method can handle RHS of the form P(t) e^{a t} for constants a, where P is a polynomial, provided that a is not a root of the characteristic equation (or you adjust by multiplying by t accordingly if it is). Here, a = 0, 9, 18, 27, 36, 45 are all achievable exponents, and 0 corresponds to a polynomial term in t; none of these exponents are roots of the characteristic equation r^2 - r + 1 = 0. Therefore, the right-hand side is indeed of a type that can be attacked with undetermined coefficients. This makes option b incorrect.

Option c: "no, because the DE is not linear." The equation y'' - y' + y = (7 e^{9t} + t)^5 is linear in y because the left-hand side is linear in y and its derivatives, and the right-hand side is a known function of t. There is no y^2, no |y|, and no product of y with its derivatives. Thus the DE is linear, and the claim that it is not linear is false. This option is incorrect.

Option d: "yes." Given the DE is linear with constant coefficients on the left and a right-hand side that, after expansion, comprises terms of the form t^m e^{a t} with a in {0, 9, 18, 27, 36, 45} and m up to 5, undetermined coefficients can be employed (with the usual caveat that for terms where a is a root of the characteristic equation one would multiply by t appropriately). Since none of these a-values are roots of r^2 - r + 1 = 0, and the RHS is a suitable combination of exponential and polynomial factors, the method of undetermined coefficients with superposition is applicable. Therefore, option d is the correct assertion.

MAP2302 L15 section 4.5

Can the method of undetermined coefficients with superposition be used to solve the DE? y″−y′+y=(7e9t+t)5 a. no, since the coefficients of the DE are not constant b. no, because the right side of the equation is not the correct type of function c. no, because the DE is not linear d. yes

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