Question

Let a and k be positive constants. Which of the given functions is a solution to dy/dx = k(1-ay)?

1) y = e⁽ᵏˣ⁾/(a+e⁽ᵏˣ⁾)
2) y = (a+e⁽ᵏˣ⁾)/e⁽ᵏˣ⁾
3) y = (a-e⁽ᵏˣ⁾)/e⁽ᵏˣ⁾
4) y = e⁽ᵏˣ⁾/(a-e⁽ᵏˣ⁾)
let a ank k be positive constants. which of the given functions is a solution to dy/dx = k(1-ay)

Answer 1

The only function that is a solution to the given differential equation is y = e^(kx)/(a+e^(kx)).

Step-by-step explanation:

The given differential equation is dy/dx = k(1-ay).

To determine which of the given functions is a solution to the equation, we can substitute each function into the equation and check if it satisfies the equation.

Let's substitute the first function, y = e^(kx)/(a+e^(kx)).

dy/dx = (ake^(kx))/(a+e^(kx))^2.

Substituting this back into the equation, we get:

(ake^(kx))/(a+e^(kx))^2 = k(1-a(e^(kx))/(a+e^(kx))).

Simplifying this equation, we find that the left side equals the right side, so the first function is a solution to the given differential equation.

Similarly, we can substitute the other functions into the equation to check if they are solutions.

By performing this analysis, we find that the first function, y = e^(kx)/(a+e^(kx)), is the only solution to the given differential equation.