Question

Where are all the calculus questions on x-mas eve?

Answer 1

\[y'' + 9y = e^{2x} + x^2 sinx\]
write a trial solution for the method of undetermined coefficients. do not determine the coefficients.
here's a problem from my calc book :P
nonhomogenous linear equation it's called..

Answer 2

for \[y_h\]
\[r^2+9=0\implies r=\pm 3i\\ y_h=c_1\cos 3x+c_2\sin 3x\]

Answer 3

Hmm, so I just need to supply a trial solution? I'll try basically that but tack on an undetermined coefficient to every x^3sinx, x^3cosx, x^2sinx, x^2cosx, etc... down to sinx and cosx along with e^3x.

I think I would never try to solve this by undetermined coefficients, but that'd be a quick first guess I suppose if I was masochistic...

Answer 4

you just multiply the guesses

Answer 5

for the nonhomogeneous part I mean

Answer 6

what guess do you make for \[y''+p(t)y'+q(t)y=g(t)\]when \[g(t)=x^2\]?

Answer 7

so now we split solutions \[y_p=y_1+y_2\]
\[w(\cos x,\sin x )=1\]
\[y_p=\cos3xy_1+\sin3xy_2\]
\[y_p=-\cos3x\int \sin 3x (e^{2x}+x^2\sin x)+\sin 3x \int \cos 3x(e^{2x}+x^2\sin x)\]

Answer 8

http://www.wolframalpha.com/input/?i=int+sin+3x+%28e%5E%7B2x%7D%2Bx%5E2sin+x%29

Answer 9

for the integral..i think this is not a friendly integral

Answer 10

you are doing variation of parameters, he wanted undetermined coefficients

Answer 11

@Jonask The question says not to solve it, just make a guess eh? Yeah that too, even though variation of parameters and transforms is far superior, I'm kinda having fun with this.

@TuringTest I'm solving yours right now let me think! =D

Answer 12

okay so with undetermined coeffecients i can get that the solution for the e^2x part is

i will make the guess
\[y_1=\frac{e^{2x}}{2^2+9}=\frac{e^{2x}}{13}\]

Answer 13

\[y=C_1e^{-p/2}\sin(\sqrt{q}/2)+C_2e^{-p/2}\cos(\sqrt{q}/2)\]

Does that look like anything useful to us? That would appear to be the answer to the homogenous, but I'm not sure how to connect that to anything other than variation of parameters and you're looking for a undetermined coefficient guess.

Answer 14

so a trial solution is just a guess looking at the general from...i havent learned this technique....seems interesting though

Answer 15

\[y''+9y=e^{2x}+x^2sinx\]
The problem I see is that if you use an Ax^2sinx as a term of an undetermined coefficient guess it will always cancel itself out.

Answer 16

Sorry, specifically any power of x will be bad. A*(x^n)*sinx

Answer 17

now people are overthinking it a bit...
@Jonask already gave us the complimentary solution to the homogeous, and so now we need the particular solution, our guess for which is entirely dependent on g(t), i.e. the stuff on the right
so we know that it's\[y=c_1\sin(3x)+c_2\sin(3x)+y_p(x)\]now if we had nothing but \[g(x)=x^2\]on the right, what would our guess for the particular be?

Answer 18

(quadratics polynomial)+(Asin x+Bcosx)

Answer 19

jus guessing!?

Answer 20

Ax^3+Bx^2+Cx+D+Esinx+Fcosx

Answer 21

@Jonask I suppose yeah? I don't know what is considered the "guess" honestly haha.

Answer 22

kanui you're very close

Answer 23

oh you already know this @shamil98 ?

Answer 24

I'm starting this section in the calculus book , i'm not too familiar with it yet.

Answer 25

well I'll help you with it if you need, but I don't want to give away the answer if you're putting it up as a challenge

Answer 26

for now it's a challenge, the solution i have atm is from the back of the book xD,i'm still looking on how to solve it .

Answer 27

Is the answer short or very long looking?

Answer 28

well like I said, the rule is simply that you want to multiply the guesses that are multiplied together on the rhs
so multiply the guess for the polynomial with that for the trig function, then know that you can add it to the guess for the exponential by superposition

Answer 29

it's a bit longer than your post above kanui

Answer 30

i think we shud leave this problem,we can leave wolfram for computation
can we get another problem,

Answer 31

thats if you are all fine with it

Answer 32

If the guess is this I'll be sad.

y=Ax^3sinx+Bx^3cosx+Cx^2sinx+Dx^2cosx+Exsinx+Fxcosx+Ge^3x+Hx^3+Ix^2+Jx+K+Lsin(3x)+M(cos3x)

Answer 33

\[y_p = Ae^{2x} + (Bx^2+ Cx+ D) \cos x +(Ex^2 + Fx + G) \sin x\]

Answer 34

not bad i think we already know A=1\13
dowable

Answer 35

y'=y