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

Explanation:

Step1: Calculate \(4y'\)

Given \(y = e^{-x}\), then \(y'=-e^{-x}\). So \(4y' = 4\times(-e^{-x})=-4e^{-x}\)

Step2: Calculate \(7y\)

Since \(y = e^{-x}\), then \(7y = 7e^{-x}\)

Step3: Calculate \(4y'+7y\)

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

Answer:

Yes