Final answer:
To solve the given initial-value problem xy' + y = ex with y(1) = 9, multiply both sides by x to find the integrating factor. Rewrite the equation as the derivative of the product xy and integrate both sides to find the general solution for y. Substitute the initial condition y(1) = 9 to find the particular solution.
Step-by-step explanation:
To solve the given initial-value problem xy' + y = ex with y(1) = 9, we'll use an integrating factor. The integrating factor is found by multiplying both sides of the equation by x. This gives us xy' + y = xex. Now we can rewrite the left side of the equation as the derivative of the product xy. The equation becomes d(xy)/dx = xex. By integrating both sides, we can find the general solution for y. Then we can substitute the initial condition y(1) = 9 to find the particular solution. Finally, the largest interval i over which the solution is defined can be determined by analyzing the behavior of the equation and its initial condition.