Respuesta :
Applying substitution, it is found that y is a solution to the differential equation for k = 1.
What is the differential equation:
It is given by:
[tex]y - \frac{y^{\prime\prime}}{4} = 5e^{4x}[/tex]
We are testing the following solution:
[tex]y = e^{-2x} + ke^{4x}[/tex]
It's derivatives are given by:
[tex]y^{\prime}(x) = -2e^{-2x} + 4ke^{4x}[/tex]
[tex]y^{\prime\prime}(x) = 4e^{-2x} + 16ke^{4x}[/tex]
Then, replacing in the differential equation:
[tex]y - \frac{y^{\prime\prime}}{4} = 5e^{4x}[/tex]
[tex]e^{-2x} + ke^{4x} - \frac{4e^{-2x} + 16ke^{4x}}{4} = 5e^{4x}[/tex]
[tex]e^{-2x} + ke^{4x} - e^{-2x} + 4ke^{4x} = 5e^{4x}[/tex]
[tex]5ke^{4x} = 5e^{4x}[/tex]
[tex]5k = 5[/tex]
[tex]k = 1[/tex]
Then, y is a solution to the differential equation for k = 1.
More can be learned about differential equations at https://brainly.com/question/14423176
