Solving this Linear Differential Equation?

[τheory]

Homework Statement

Find the general solution to this differential equation:
\frac{dy}{dx}+18x^{17}y=x^{18}
Use the variable I for replacing \int e^{x^{18}} dx

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

My problem is using the variable I for replacing the \int e^{x^{{18}}} dx. How am I suppose to do this with x^{18} inside the integral?

 

[vela]

You need to group characters using { } in LaTeX.

Homework Statement

Find the general solution to this differential equation:
\frac{dy}{dx}+18x^{17}y=x^{18}Use the variable I for replacing \int e^{x^{18}}\,dx

2. The attempt at a solution
I've solved the differential equation using integrating factors and obtained the following result:
y=\frac{\int e^{x^{18}}x^{18}\,dx + C}{e^{x^{18}}}My problem is using the variable I for replacing the \int e^{x^{18}}\,dx. How am I suppose to do this with x^18 inside the integral?

You can't. You want to integrate by parts now.

 

[flyingpig]

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 e^{x^18}, why do you have \int e^{x^18} dx?

 

[τheory]

Okay well I did try using integration by parts after I obtained the general solution, but didn't get far as I got this:
\int e^{x^{18}}x^{18}dx

u = x^{18}
du = 18x^{17} dx
dv = e^{x^{18}} dv
v = \int e^{x^{18}} dv

uv - \int v du

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

At this point, how am I suppose to utilize the substitution of \int e^{x^{18}}dx with I? 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 I for every \int e^{x^{18}}dx 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.

 

[vela]

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