Final answer:
To find the value of y0 for the differential equation y' - y = 5 - 5sin(t), one must use the method of integrating factors and the given initial condition to solve for y(t) and then determine y0.
Step-by-step explanation:
To find the value of y0 for which the solution of the initial value problem y' - y = 5 - 5sin(t) holds true, we will first need to solve the differential equation. We can employ the method of integrating factors to do this.
The integrating factor, μ(t), is obtained by taking the exponential of the integral of -1, which is simply e^{-t}. Multiplying both sides of the differential equation by this integrating factor transforms the left side into the derivative of yμ(t). Now, we can integrate both sides with respect to t, and then divide by μ(t), to solve for y.
After finding the general solution of y(t), we can use the initial condition provided to solve for y0. Substituting t = 0 and the y(0) value we are given into the solution gives us the specific value of y0.