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 } )
(b) find ( 4 y ^ { prime } ), ( 7 y ), and ( 4 y ^ { prime } + 7 y ) for ( y = e ^ { - x } + e ^ { - ( 7 / 4 ) x } ).
( 4 y ^ { prime } = - 4 e ^ { - x } - 7 e ^ { - ( 7 / 4 ) x } )
( 7 y = 7 e ^ { - x } + 7 e ^ { - ( 7 / 4 ) x } )
( 4 y ^ { prime } + 7 y = 3 e ^ { - 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

Explanation:

Step1: Substitute \(4y'\), \(7y\) and \(4y' + 7y\) into the differential equation

We are given the differential equation \(4y'+7y = 3e^{-x}\).
For the function \(y=e^{-x}+e^{-(7/4)x}\), we have found that \(4y'=-4e^{-x}-7e^{-(7/4)x}\), \(7y = 7e^{-x}+7e^{-(7/4)x}\), and \(4y'+7y=3e^{-x}\).

Step2: Check if the left - hand side equals the right - hand side

The left - hand side of the differential equation \(4y' + 7y\) for \(y = e^{-x}+e^{-(7/4)x}\) is \(3e^{-x}\), and the right - hand side of the differential equation \(4y' + 7y=3e^{-x}\) is also \(3e^{-x}\).

Answer:

Yes