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 } )
is the function ( y = e ^ { - x } + e ^ { - ( 7 / 4 ) x } ) a solution of ( 4 y ^ { prime } + 7 y = 3 e ^ { - x } )? choose the correct answer below.
yes
no
(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 } =)
( 7 y =)
( 4 y ^ { prime } + 7 y =)

Explanation:

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

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

Step2: Calculate \(4y^\prime\)

Multiply \(y^\prime\) by \(4\): \(4y^\prime=-4e^{-x}-7Ce^{-(7/4)x}\)

Step3: Calculate \(7y\)

Multiply \(y\) by \(7\): \(7y = 7e^{-x}+7Ce^{-(7/4)x}\)

Step4: Calculate \(4y^\prime + 7y\)

Add \(4y^\prime\) and \(7y\):

$$
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$$

Answer:

\(4y^\prime=-4e^{-x}-7Ce^{-(7/4)x}\)

\(7y = 7e^{-x}+7Ce^{-(7/4)x}\)

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