Integrating a Problematic Exponential: Solving 1st Order Linear ODE with e^(x^2)

[DryRun]

Homework Statement
dy/dx + y/x = e^(x^2)
Express y in terms of x and arbitrary constant.

The attempt at a solution
It is in the standard 1st order linear ODE form.

P(x) = 1/x
Q(x) = e^(x^2)

u(x) = x (after calculation)

So, d(uy)/dx = uQ

d(uy)/dx = x.e^(x^2)
I have to integrate both sides w.r.t.x

Finding the R.H.S is problematic though. As it seems infinite, from what I've understood from my calculations.

Integral of x.e^(x^2) (done by partial integration)
Let U = x, so dU/dx = 1
Let dV = e^(x^2), so V = [e^(x^2)]/2x (is this correct?)

xy = x.[e^(x^2)]/2x - integral of [e^(x^2)]/2x.(1).dx

Then i have to integrate the R.H.S again and again and again, as i can't get rid of e^(x^2) with a multiple of x always in the denominator.

Any advice?

 

[Quinzio]

$$y'+a(x)y = b(x)$$
$$y=e^{-\int a(x)dx} \int e^{\int a(x)dx}b(x)dx$$

 

[DryRun]

Hi Quinzio

I know about that formula. I used it in the first place, and the spot where I'm stuck is due to b(x) in your equation being e^(x^2). The power of e contains x which is giving me quite some trouble with this problem.

 

[Quinzio]

Ok, but you got an x which is its derivative (of $x^2$).
It goes:
$$a(x)=1/x$$
$$\int a(x)dx=ln|x|$$
$$e^{\int a(x)dx}=|x|$$
$$\int e^{\int a(x)dx}b(x)dx=|x|e^{x^2}=\frac{1}{2}e^{x^2}*sgn(x)$$

 

[DryRun]

I did some research and it appears that the integral of e^(x^2) is infinity and cannot be expressed in normal terms.

However, it is curious that my calculus notes give this as the answer to this problem:

y= [e^(x^2)]/2x + A/x

 

[Quinzio]

Your notes show the right answer.
I just wrote a part of it, what I thought was the harder.
The answer in your notes is for x>=0 only, anyway.

 

[DryRun]

This is the last line where I've been able to reach:

xy = x.[e^(x^2)]/2x - integral of [e^(x^2)]/2x.(1).dx

I'm stuck at the R.H.S. where i need to integrate [e^(x^2)]/2x

I realize that i know nothing about the sign function. If that is a requirement to solve this problem, then i won't be able to do this.

Is there any easy guide that you could possibly recommend to learn about the sign function?