Question

I have a questions about DE

Answer 1

I know how to solve differential equations in a form of:

\(ay''+by'+cy=0\)

but, how would I got about solving something like:

\(ay''+by'+cy=p(x)\)

can anyone tell me,
(if it is not too far beyond the regular ay''+by'+cy=0)?

Answer 2

well...there is no standard way to solve the second equation, it has to be solved on a case by case basis using a mix of clever tricks such as "complexification" and guess work like "undetermined co-coefficients".... dont worry about this until you are actually taught these :)

Answer 3

what if I had the following?

\(y''+4y'+3y=x\)

Answer 4

do I still set?

\(r^2+4r+3=x\)

Answer 5

NO! remember that the charateristic polynomial works for only for homogeneus case, i.e p(x)=0

for this eq, you have to guess that the soulution will be of the form y= Ax+B (where A and B are unkown). substitute this in the DE, and compare LHS and RHS to find A and B

Answer 6

0+4A+3Ax+B3=Ax+B ?

Answer 7

oh, the left is just x, not p(x)

Answer 8

0+4A+3Ax+B3=x ?

Answer 9

re arrange this way
(3A)x + (4A+3B) = x+ 0

now compare co-efficeints of x
and compare the constant term{0 and (4A+3B)}

Answer 10

One cute way that's pretty general but really is just undetermined coefficients is to factor out the derivative since it's just a linear operator:

\[(aD^2+bD+cI)y(x)=p(x)\]
Then depending on what \(p(x)\) is, like specifically if it's sines, cosines, or exponentials which are basically closed under differentiation, then you can easily find the derivative operators in terms of finite matrices, and then just invert the matrix \((aD^2+bD+cI)\) to solve for your function:

\[y(x) = (aD^2+bD+cI)^{-1} p(x)\]

I could give an example depending on how comfortable you are with matrices, like if you've touched 2x2 matrices those are probably the easiest to work with, so you could try solving this differential equation:

\[3y''-7y'+8y=2\cos (4x)\]

Answer 11

undetermined coefficients..... my fears became true, this is beyond my current knowledge for me to learn... sorry for wasting time-:(

Answer 12

I got to go, thank you for replying:)

Answer 13

Hahaha it's not that bad, undetermined coefficients is really not a big deal.

Answer 14

Yeah, come ask some fun differential equations questions any time, I'll try to say less scary stuff :P

Answer 15

Pssst @zepdrix are you trying to solve that diffeq I made up or are you just afk

Answer 16

lol afk soz :)

Answer 17

I BARELY got through matrix and linear :o(
I'm gonna have to revisit that material hardcore...

Answer 18

haha I've really started digging into the linear algebra lately so I kinda keep trying to bring it up everywhere xD