We want to solve the given differential equation. We will see that the correct option is A:
[tex]y(x) = 4*e^{4*x}[/tex]
We know that:
[tex]\frac{dy}{dx} = 4y\\\\y(0) = 4[/tex]
For a given function y(x).
Because the derivate is a constant times the original function, we know that the original function must be something like:
[tex]y(x) = A*e^{4x}[/tex]
Where A is a constant. Notice that if we derive the above function we get:
[tex]y'(x) = 4*A*e^{4x} = 4*y(x)[/tex]
Now we use the initial value restriction, we know that when we evaluate this in x = 0, we should get 4, then:
[tex]y(0) = 4 = A*e^{4*0} = A[/tex]
Then c = 4, then the equation is:
[tex]y(x) = 4*e^{4*x}[/tex]
So the correct option is A.
If you want to learn more, you can read:
https://brainly.com/question/353770
