Final answer:
To solve the differential equation x²y' + xy = 1, an integrating factor of |x| is used to obtain the general solution y = (ln|x|)/|x| + C/|x|. Using the initial condition y(9) = 10, the constant C is found to be 90 - ln(9). The particular solution for the initial condition given is y = (ln|x|)/|x| + (90 - ln(9))/|x|.
Step-by-step explanation:
The differential equation given is x²y' + xy = 1. This is a first-order linear differential equation. To solve it, the first thing to do is to rewrite the equation in standard form. Dividing through by x², we have y' + (1/x)y = 1/x². Recognizing that the term (1/x) is the derivative of ln|x|, we can see that this is a linear differential equation with integrating factor e²(ln|x|) or |x|. Multiplying through by the integrating factor, we get |x|y' + y|x| = |x|/x² or (|x|y)' = 1/x. Integrating both sides gives |x|y = ln|x| + C, so y = (ln|x|)/|x| + C/|x|.
To find the specific solution that satisfies the initial condition y(9) = 10, substitute x = 9 and y = 10 into the general solution to solve for C. Therefore, 10 = (ln|9|)/|9| + C/|9|, which gives C = 90 - ln(9). The particular solution is y = (ln|x|)/|x| + (90 - ln(9))/|x|.