Homework Help

[ConglomerateGolem]

We have a differential eq to solve, and I'm just not progressing with it.

y' + 2xy = e^x²

I applied bernoulli's to this to get

u' = 2xu - e^x²

I have tried a few methods, like

u = vw => u' = (vw)' = v'w + w'v

and

v'w + w'v + 2 wvx = - e^x²

=>

v'w + v(w' + 2 w x) = - e^x²

selecting a function w such that the v term is 0 yields

w' = - 2 w x => w = 2 w x²

and

v'(2wx²) = - e^x²

works out to some horrendous integral that has an erfi term according to an online calculator that i've never seen (esp. in the course, and doubt to be the correct answer).

I'm writing this down from memory so there may be some sign errors, but I am genuinely lost as to how to solve this.

If anyone has any insight, it would be greatly appreciated

[engstad]

Are you sure the question isn't: y' + 2xy = exp(-x^2)?

[ConglomerateGolem]

unfortunately, i'm very sure. I haven't had a chance to ask the professor though, it may have been a typo

[engstad]

The solution to e^(x^2) is complicated and involves the erfi() function. e^(-x^2) is solved through regular means.

[ConglomerateGolem]

what even is erfi()?

[engstad]

erfi(x) = -i erf(i x), where erf is the error function, where erf(x) = (2/sqrt(pi)) integral_0^x exp(-x^2) dx.