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Solving this Linear Differential Equation?

  1. Nov 23, 2011 #1
    1. The problem statement, all variables and given/known data
    Find the general solution to this differential equation:
    [itex]\frac{dy}{dx}+18x^{17}y=x^{18}[/itex]
    Use the variable [itex]I[/itex] for replacing [itex]\int e^{x^{18}} dx[/itex]

    2. The attempt at a solution
    I've solved the differential equation using integrating factors and obtained the following result:
    [itex]y=\frac{\int e^{x^{18}}x^{18} dx + C}{e^{x^{18}}}[/itex]

    My problem is using the variable [itex]I[/itex] for replacing the [itex]\int e^{x^{{18}}} dx[/itex]. How am I suppose to do this with [itex]x^{18}[/itex] inside the integral?
     
    Last edited: Nov 24, 2011
  2. jcsd
  3. Nov 24, 2011 #2

    vela

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    You need to group characters using { } in LaTeX.
    You can't. You want to integrate by parts now.
     
  4. Nov 24, 2011 #3
    Are you familiar with the product rule?

    (f*g)' = f'g + g'f

    I am not following what you are doing at all...

    You already got the integrating factor [tex]e^{x^18}[/tex], why do you have [tex]\int e^{x^18} dx[/tex]?
     
  5. Nov 24, 2011 #4
    Okay well I did try using integration by parts after I obtained the general solution, but didn't get far as I got this:
    [itex]\int e^{x^{18}}x^{18}dx[/itex]

    [itex]u = x^{18}[/itex]
    [itex]du = 18x^{17} dx[/itex]
    [itex]dv = e^{x^{18}} dv[/itex]
    [itex]v = \int e^{x^{18}} dv[/itex]

    [itex]uv - \int v du[/itex]

    [itex]x^{18}\int e^{x^{18}} dv - \int[ \int e^{x^{18}} dv] 18x^{17} dx[/itex]

    At this point, how am I suppose to utilize the substitution of [itex]\int e^{x^{18}}dx[/itex] with [itex]I[/itex]? In other words, how do I input the general solution into the website without inputting integrals? Since the website doesn't recognize integral signs, which is why it's asking me to use [itex]I[/itex] for every [itex]\int e^{x^{18}}dx[/itex] I encounter.

    To the flyingpig's question, I need to use the variable "I" because it says to do so in the online problem that I'm doing.
     
  6. Nov 24, 2011 #5

    vela

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    Try using u=x and [itex]dv=x^{17}e^{x^{18}}\,dx[/itex].
     
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