Practice... (made up an example).

Question

anonymous · Answer

$\large y'+yx=3$

anonymous · Answer

$\large \displaystyle \int_{}^{}x~dx~=~\frac{x^2}{2}$   (of course, we omit the +c)

So the integrating factor $\large e^{\rm H(x)}$ is in this case $\large e^{x^2/2}$.
And now we multiply everything by this integrating factor.

$\large y'~e^{x^2/2}+y~e^{x^2/2}~x=3~e^{x^2/2}$

now the simple use of the product rule use (( which I found very clever
as I read... very cool to come up with "integrating factor" in such equations
as   y' + p(x) y = q(x)    ))

$\large \displaystyle \frac{d}{dx}\left[y~e^{x^2/2}\right]=3~e^{x^2/2}$
Integrating both sides

$\large \displaystyle y~e^{x^2/2}=\int 3~e^{x^2/2}~dx$
No closed form.... BUT....

anonymous · Answer

$\displaystyle \large e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}   $

$\displaystyle \large e^{x^2/2}=\sum_{n=0}^{\infty}\frac{x^{2n}}{2^n~n!}   $

$\displaystyle \large \int e^{x^2/2}~ dx=\int \sum_{n=0}^{\infty}\frac{x^{2n}}{2^n~n!}~dx   $

$\displaystyle \large \int e^{x^2/2}~ dx=\sum_{n=0}^{\infty}\frac{x^{2n+1}}{(2n+1)~2^n~n!}   $

$\displaystyle \large \int 3e^{x^2/2}~ dx=3\sum_{n=0}^{\infty}\frac{x^{2n+1}}{(2n+1)~2^n~n!}   $

anonymous · Answer

$\displaystyle \Large y=\frac{\displaystyle 3\sum_{n=0}^{\infty}\frac{x^{2n+1}}{(2n+1)~2^n~n!}+C }{~~~~~~~~~~~e^{x^2/2}~~~~~~~~~}  $

anonymous · Answer

@radar @wio @triciaal 

Is this solution that I came up is super weird?
Or is my example just impossible?
Or did I do something totally incorrect?

anonymous · Answer

@hamidic

anonymous · Answer

I personally don't see any errors in your work. However, I've never taken a formal diff eq course, I just learned it on my own, so not sure if there are any special conventions about this kind of stuff. Other than that, nothing wrong with the calc itself.

anonymous · Answer

I really want to say that my example is very bad, because I can't get an elementary function when I integrate e^(x^2/2), that is why y'+yx=3.

If I had a teacher.... but I read just a couple hours ago in my calculus book.... tnx for checking, will see what others say if they see this.