Hi guys I'm trying to get the exact solution of -au''+bu'=x^2, really stuck any help appreciated

Question

zepdrix · Answer

Were you able to find the Complementary Solution at least?

I can't help you with the other part :C I just learned about these today, I need to go over them still lol.

anonymous · Answer

Yeah went through the central differencing and truncation error no problem, but I just seem to be going nowhere with the exact solution

anonymous · Answer

Okay so what is a function where if you take the derivative and subtract the second derivative, you get $x^2$?

anonymous · Answer

It's probably something like $\large x^n$ right?

anonymous · Answer

So let $u = cx^n$ and let's see if we can't find out what $c$ and $n$ are.  You following @TKelly

anonymous · Answer

Yeah sorry just cooking my dinner right now, trying to multitask

anonymous · Answer

$$bu^\prime = bncx^{n-1}$$$$-au^{\prime \prime} = -a n(n-1)x^{n-2}$$$$x^2=bncx^{n-1}-an(n-1)x^{n-2}$$Something like this...?

anonymous · Answer

Actually I messed up.

anonymous · Answer

oh and the boundary conditions are easy u(0)=u(1)=0 I think I've got it from here thanks wio

anonymous · Answer

Okay let $$u = Ax^2+Bx+C$$$$u^\prime = 2Ax+B$$$$u^{\prime\prime} = 2A$$

anonymous · Answer

$$x^2 = -a2A + b(2Ax+B)$$Maybe I shouldn't have reused a, but anyway, something like this.  @TKelly

anonymous · Answer

Hmmm this might not work either!!  Woops!

anonymous · Answer

Anyway I hope I helped somewhat.