Solving this Linear Differential Equation?

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?

Last edited:

vela
Staff Emeritus
Homework Helper
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.

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 $$\int e^{x^18} dx$$?

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