Question

check whether each of the following functions is a solution of the differential equation ( 4 y^{prime}+7 y = 3 e^{-x} )(a) ( y = e^{-x} )(b) ( y = e^{-x}+e^{-(7 / 4) x} )(c) ( y = e^{-x}+c e^{-(7 / 4) x} )(c) find ( 4 y^{prime}, 7 y ), and ( 4 y^{prime}+7 y ) for ( y = e^{-x}+c e^{-(7 / 4) x} ).( 4 y^{prime}=-4 e^{-x}-7 c e^{-(7 / 4) x} )( 7 y = 7 e^{-x}+7 c e^{-(7 / 4) x} )( 4 y^{prime}+7 y = 3 e^{-x} )is the function ( y = e^{-x}+c e^{-(7 / 4) x} ) a solution of ( 4 y^{prime}+7 y = 3 e^{-x} )? choose the correct answer below.noyes

Explanation:

Step1: Recall the definition of a solution to a differential equation

A function \(y = f(x)\) is a solution of the differential equation \(4y'+7y = 3e^{-x}\) if when we substitute \(y\) and \(y'\) into the left - hand side of the differential equation (\(4y'+7y\)), we get the right - hand side (\(3e^{-x}\)).

Step2: Analyze the given results

We are given that for \(y=e^{-x}+Ce^{-(7/4)x}\), \(4y'=-4e^{-x}-7Ce^{-(7/4)x}\) and \(7y = 7e^{-x}+7Ce^{-(7/4)x}\). Then, when we calculate \(4y'+7y\):

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Answer:

Yes