Question

check whether each of the following functions is a solution of the differential equation (4y+7y = 3e^{-x}) (a) (y = e^{-x}) (b) (y=e^{-x}+e^{-(7/4)x}) (c) (y = e^{-x}+ce^{-(7/4)x}) is the function (y = e^{-x}) a solution of (4y+7y = 3e^{-x})? choose the correct answer below. no yes (b) find (4y), (7y), and (4y+7y) for (y = e^{-x}+e^{-(7/4)x}). (4y=) (7y=) (4y+7y=)

Explanation:

Step1: Find the derivative of \(y = e^{-x}+e^{-(7/4)x}\)

Using the chain rule \((e^{ax})^\prime = ae^{ax}\), we have \(y^\prime=-e^{-x}-\frac{7}{4}e^{-(7/4)x}\). Then \(4y^\prime = 4(-e^{-x}-\frac{7}{4}e^{-(7/4)x})=-4e^{-x}-7e^{-(7/4)x}\)

Step2: Calculate \(7y\)

Substitute \(y = e^{-x}+e^{-(7/4)x}\) into \(7y\), we get \(7y = 7e^{-x}+7e^{-(7/4)x}\)

Step3: Calculate \(4y^\prime + 7y\)

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Answer:

\(4y^\prime=-4e^{-x}-7e^{-(7/4)x}\), \(7y = 7e^{-x}+7e^{-(7/4)x}\), \(4y^\prime + 7y=3e^{-x}\)