Final answer:
To solve the differential equation y' = 15 - 8y with the initial condition y'(0) = 0, one can use separation of variables, integrate both sides, and apply the initial condition to find the constant of integration, leading to the final particular solution for y(t).
Step-by-step explanation:
To solve the differential equation y' = 15 - 8y with the initial condition y'(0) = 0, we first recognize this as a first-order linear ordinary differential equation. To solve it, we can use the method of separation of variables or an integrating factor. In this case, we will apply the separation of variables.
Separating variables involves rearranging the equation to isolate y and its derivatives on one side, and t on the other. We integrate both sides of the equation to find the general solution. To find the particular solution that fits the initial condition, we will use the given information to determine the constant of integration.
Upon solving the differential equation, we obtain a general solution of the form:
y(t) = A exp(-8t) + 15/8
where A is an unknown constant. We use the initial condition to solve for A. Since y'(0) = 0, we substitute t = 0 into the derivative of our solution and set it equal to zero to solve for A. This will yield the particular solution that satisfies both the differential equation and the initial condition.
Once we determine A, we simply substitute it back into the general solution to get the final answer for y(t).