Final answer:
To solve the initial value problem y'' - 3y' + 2y = 0 with y(0) = -7/3, we can use the method of characteristic equation and find the general solution. Then, we can use the initial condition to determine the particular solution. The solution is y = e^x - 3e^(2x).
Step-by-step explanation:
To solve the initial value problem y'' - 3y' + 2y = 0 with y(0) = -7/3, we can use the method of characteristic equation.:
Step 1: Find the characteristic equation by assuming y = e^(rx). The characteristic equation is r^2 - 3r + 2 = 0.
Step 2: Solve the quadratic equation using factoring or quadratic formula. The roots are r = 1 and r = 2.
Step 3: The general solution is given by y = C1 * e^(r1x) + C2 * e^(r2x). Substituting the values of r1 = 1 and r2 = 2 and using the initial condition, we can determine the particular solution.
Step 4: Plug in the values of x = 0 and y = -7/3 to solve for the constants C1 and C2.
Therefore, the solution to the initial value problem is y = e^x - 3e^(2x).