SOLUTION: Solve y′+3x^(−1)y=x^(8),y(1)=9. (a) Identify the integrating factor, α(x). α(x) = (b) Find the general solution. y(x)= Note: Use C for an arbitrary constant. (c) So

Question 1184442: Solve y′+3x^(−1)y=x^(8),y(1)=9.
(a) Identify the integrating factor, α(x).
α(x) =
(b) Find the general solution.
y(x)=
Note: Use C for an arbitrary constant.
(c) Solve the initial value problem y(1)=9.
y(x)=
Found 2 solutions by Edwin McCravy, robertb:
Answer by Edwin McCravy(20064) (Show Source): You can put this solution on YOUR website!

There is no need for an integrating factor when the variables are separatable. Finish integrating and put an arbitrary constant on the end. That is the general solution. Now substitute which means we substitute x=1 and y=9 It would look neater if you factored out 1/9, and you can write y(x) for y. Edwin

Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
The "procedure" shown by Edwin McCravy is basically correct. However, the problem was oversimplified when, instead of considering the D.E.

, he unwittingly started the solution with , turning it into a different D.E. that is easier to solve. (A separable D.E.)
(a) The integrating factor is .

(b) Using this integrating factor, the general solution will be .

(c) The solution to the IVP with y(1) = 9 is . As a check, you may plug this equation into the D.E. and verify that it is the solution to the IVP.
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