Final answer:
To solve the initial value problem y'' + y' = x with the initial conditions y(0) = 1 and y'(0) = 0, we can use the method of undetermined coefficients. The solution to the initial value problem is y = 2 + e^{-t} - x.
Step-by-step explanation:
To solve the initial value problem y'' + y' = x with the initial conditions y(0) = 1 and y'(0) = 0, we can use the method of undetermined coefficients. First, solve the homogeneous part of the equation: y'' + y' = 0. The characteristic equation is r^2 + r = 0, which gives us the solutions r = 0 and r = -1. Therefore, the general solution for the homogeneous part is y_h = c_1 + c_2e^{-t}, where c_1 and c_2 are constants.
Next, we find a particular solution y_p for the non-homogeneous part of the equation. Since the right hand side of the equation is x, we guess a particular solution of the form y_p = ax + b, where a and b are constants. Substituting this into the equation, we get 2a + a - ax + b = x, which implies a = -1 and b = 0.
Therefore, the particular solution is y_p = -x. Combining the general solution for the homogeneous part and the particular solution, we have y = y_h + y_p = c_1 + c_2e^{-t} - x. Applying the initial conditions, we find the values of the constants c_1 and c_2 to be c_1 = 2 and c_2 = -1. Therefore, the solution to the initial value problem is y = 2 + e^{-t} - x.