Final answer:
To solve the given initial-value problem, we can use the method of undetermined coefficients. We can find the particular solution and the general solution by assuming a particular solution of the form y = Ax + B and solving for the coefficients using the given differential equation and initial conditions. The specific solution to the initial-value problem is y = e^(2/3 * x) + 1/3 * x.
Step-by-step explanation:
To solve the given initial-value problem, we can use the method of undetermined coefficients. We assume a particular solution of the form y = Ax + B for the differential equation. By substituting this assumed solution into the equation and solving for A and B, we can find the particular solution. Finally, we can add the particular solution to the complementary solution to obtain the general solution.
Given the initial conditions y(0) = 0 and y'(0) = 1, we can substitute these values into the general solution to find the specific solution to the initial-value problem.
In this case, the particular solution is y = 1/3 * x, and the complementary solution is y = e^(2/3 * x). Therefore, the general solution is y = e^(2/3 * x) + 1/3 * x. Substituting the initial conditions, we find the specific solution: y = e^(2/3 * x) + 1/3 * x.