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

y″−2y′+3y=2sint−e6tt2+
1

t

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″−2y′+3y=2sint−e6tt2+
1

t

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

To tackle the question, I’ll parse the problem and the options, then examine what each choice claims and whether it aligns with the method of undetermined coefficients with superposition.
Option a: 'no, since the coefficients of the DE are not constant.' The differential equation given has y'' − 2y' + 3y on the left, where the coefficients of y'' and y' are 1 and −2, and the coefficient of y is 3; these are constant. Therefore this reason is incorrect because constant-coefficient linear DEs do exist, and the method in question typically applies in such cases. This option wrongly asserts a non-constant coefficient situation.
Option d: 'yes.' This asserts that the method of undetermined coefficients with superposition can be used. For that method to be valid, the ODE should be linear with constant coefficients and the forcing term should be a linear combination of terms of recognizable forms (polynomials, exponentials, sines, cosines, or products of those with polynomials). If the forcing term includes a term like 1/t, that particular form is not among the standard solvable-by-ansatz types, and the undetermined coefficients method does not apply straightforwardly. The validity depends on the exact right-hand side; if 1/t is present, this option would be questionable.
Option c: 'no, because the DE is not linear.' The equation y'' − 2y' + 3y = … is linear in y and its derivatives since the unknown function and its derivatives appear only to the first power and are not multiplied by each other. This option is incorrect because the DE, as written, is linear.
Option b: 'no, because the right side of the equation is not the correct type of function.' If the forcing term includes a nonstandard form such as 1/t (or any term not expressible as a polynomial in t, an exponential e^{αt} times a polynomial, or a sine/cosine (and their multiples by polynomials), then the standard undetermined coefficients approach does not apply directly. In that case, one would typically use variation of parameters or another method rather than the undetermined coefficients with superposition. Hence this option identifies a potential mismatch between the forcing term’s form and the method’s applicability.
In summary, option a errs by mischaracterizing the coefficient structure, option c wrongly claims nonlinearity, option d asserts applicability without checking the forcing term’s form, and option b points to a mismatch in the forcing term’s form that would prevent using the undetermined coefficients method in its standard form. Given the common criteria for the method’s applicability, the most defensible critique among these is that the right-hand side is not of the standard type required for the undetermined coefficients approach, which aligns with the reasoning in option b.

MAP2302 L15 section 4.5

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