Question

Let y=f(x) be a solution of the differential equation y′ =ky, where k is a constant. If f(0)=8 and f(4)=2, which of the following is an expression for f(x)?

a) f(x)=8e⁻ˣ/²
b) f(x)=8e⁻²ˣ
c) f(x)=8eˣ/²
d) f(x)=8e²ˣ

Answer 1

The expression for f(x) is f(x) = 8e^(-x/2).

Step-by-step explanation:

The differential equation given is y' = ky, where k is a constant. To find the expression for f(x), we need to solve the differential equation using the initial conditions f(0) = 8 and f(4) = 2.

Integrating both sides of the equation, we get ln|y| = kx + C, where C is the constant of integration. Substituting the initial condition f(0) = 8, we get ln|8| = k(0) + C, which gives us C = ln|8|.

Now, substituting the initial condition f(4) = 2, we get ln|2| = k(4) + ln|8|. Solving for k, we find k = (ln|2| - ln|8|)/4 = ln(1/4).

Therefore, the expression for f(x) is f(x) = 8e^(ln(1/4)x) = 8(1/4)^x/2 = 8e^(-x/2).