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

Algebra ->  Expressions-with-variables -> 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      Log On


   

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(20063) About Me  (Show Source):
You can put this solution on YOUR website!
There is no need for an integrating factor when the variables are
separatable.

%22y%27%22%2B3x%5E%28-1%29y%22%22=%22%22x%5E8%22%2C%22y%281%29=9

%22y%27%22%2B3%2Fx%22%22=%22%22x%5E8

%22y%27%22%22%22=%22%22x%5E8-3%2Fx

dy%2Fdx%22%22=%22%22x%5E8-3%2Fx

dy%22%22=%22%22x%5E8%2Adx-expr%283%2Fx%29dx

int%28dy%5E%22%22%29%22%22=%22%22int%28x%5E8%2Adx%29%22%22-%22%22int%28expr%283%2Fx%29dx%29

y%22%22=%22%22x%5E9%2F9%5E%22%22%22%22-%22%223int%28dx%2Fx%29

Finish integrating and put an arbitrary constant on the end.

y%28x%29%22%22=%22%22x%5E9%2F9%5E%22%22%22%22-%22%223%2Aln%28x%29%22%22%2B%22%22C

That is the general solution.

Now substitute y%281%29=9 which means we substitute x=1 and y=9

9%22%22=%22%221%5E9%2F9%5E%22%22%22%22-%22%223%2Aln%281%29%22%22%2B%22%22C

9%22%22=%22%221%2F9%22%22-%22%223%2A0%22%22%2B%22%22C

9%22%22=%22%221%2F9%22%22%2B%22%22C

9-1%2F9%22%22=%22%22C

81%2F9-1%2F9%22%22=%22%22C

80%2F9%22%22=%22%22C

y%22%22=%22%22x%5E9%2F9%5E%22%22%22%22-%22%223%2Aln%28x%29%22%22%2B%22%2280%2F9

It would look neater if you factored out 1/9, and you can write
y(x) for y.

y%28x%29+=+expr%281%2F9%29%28x%5E9+-+27%2Aln%28x%29+%2B+80%29

Edwin

Answer by robertb(5830) About Me  (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.

%22y%27%22%2B3%281%2Fx%29y=x%5E8, he unwittingly started the solution with %22y%27%22%2B3%2Fx=x%5E8, turning it into a different D.E. that is easier to solve. (A separable D.E.)
(a) The integrating factor is alpha%28x%29+=+e%5E%283int%281%2Fx%2Cdx%29%29+=+x%5E3.

(b) Using this integrating factor, the general solution will be y+=+x%5E9%2F12+%2B+C%2Fx%5E3.

(c) The solution to the IVP with y(1) = 9 is y+=+x%5E9%2F12+%2B+107%2F%2812x%5E3%29. As a check, you may plug this equation into the D.E. and verify that it is the solution to the IVP.