Question

Show that the function: y f(x) = 4e3* -3e is a solution to the differential equation: y"-2y'-3y 0

Answer 1

Answer: The verification is done below.

Step-by-step explanation: We are given to show that the function
`f(x)=4e^(3x)-3e^(-x)` is a solution to the following differential equation :

`y^(\prime\prime)-y^(\prime)-3y=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)`

If y = f(x), then we see that

`y^\prime=(d)/(dx)f(x)=(d)/(dx)(4e^(3x)-3e^(-x))=12e^(3x)+3e^(-x),\\\\\\y^(\prime\prime)=(d)/(dx)(12e^(3x)+3e^(-x))=36e^(3x)-3e^(-x).`

Therefore, we get

`L.H.S.\\\\\\=y^(\prime\prime)-2y^(\prime)-3y\\\\\\=(36e^(3x)-3e^(-x))-2(12e^(3x)+3e^(-x))-3(4e^(3x)-3e^(-x))\\\\\\=36e^(3x)-3e^(-x)-24e^(3x)+6e^(-x)-12e^(3x)+9e^(-x)\\\\=0\\\\=R.H.S.`

Thus, the function
`f(x)=4e^(3x)-3e^(-x)` is a solution to the given differential equation.

Hence showed.