y"+y=csc^3xcotx
VARIATION OF PARAMETERS :))
i have an answer but the problem is the process in getting the answer. the wolframalpha has an incorrect answer :D
@TuringTest :D

- anonymous

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- TuringTest

\[y_c=c_1\cos x+c_2\cos x\]got the wronskian yet?

- anonymous

y=C1cosx+C2sinx+1/6cotxcscx that's my answer but i dont know where does 1/6cotxcscx came from. :)

- TuringTest

whoa one of those should be sin

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## More answers

- anonymous

yea
the wronkian is 1

- TuringTest

let me see...

- anonymous

i got the answer from the book but my book has no solution. tsk

- TuringTest

\[y_p-\cos x\int \csc^2x\cot xdx+\sin x\int\csc^4 x\cos^2 xdx\]unless I made a mistake...

- TuringTest

yp=

- TuringTest

first integral is easy, the next will clearly require some trig manipulation

- anonymous

yea. i find hard time in the next integration. can you help me with this? :)

- TuringTest

I'm trying to think of the smart way to do it...

- anonymous

HAHA. whoa. i hope your smart brain can help with this. :)

- TuringTest

across could do this in her sleep, yet she remains silent....

- anonymous

HAHAHA. she's just viewing my problem :(( urgghh.

- TuringTest

oh we can rewrite this as\[\int\csc^2x\cot^2xdx\]as well, so it's doable by u-sub

- TuringTest

\[u=\cot x\implies du=-\csc^2x dx\]

- TuringTest

\[-\int u^2du=-\frac13\cot^3x+C\]

- TuringTest

you can do the other integral, right?

- anonymous

nope. HAHA. can you finish it? :)

- TuringTest

try the exact same u-sub!\[\int\csc^2x\cot x dx\]\[u=\cot x\implies du=-\csc^2x\]

- anonymous

-1/2cot^2x+C am i right?

- TuringTest

yep

- TuringTest

so now plug all that stuff into the formula

- anonymous

so the yp= -1/3cot^3x-1/2cot^2x? and the answer will be ???

- TuringTest

you forgot to multiply by y1 and y2 repectively

- TuringTest

yp= -1/3sinx*cot^3x+1/2cosx*cot^2x
looks like they did some fancy trig identities from there maybe

- TuringTest

I gotta eat breakfast, good luck simplifying!

- anonymous

okay. thanks. :) happy eating

- anonymous

the answer will be yp= 1/6 cotxcscx????

- TuringTest

hopefully

- anonymous

awwwweee.??? so my solution will be wronskian, integration then yp and lastly y? right?

- TuringTest

yep
1)find complimentary solution yc
2)find wronskian
3)apply formula for variation of parameters to get the particular solution yp
4)add the complimentary and particular to find y=yc+yp

- anonymous

can you solve my another problem that i linked you? :) the reduction of order i have the same problem with this. i mean im figuring out the process in getting the same answer :)

- TuringTest

I'm pretty much out the door to class...
you can post and maybe I'll help, but I'm not the only one here who can do DE's

- anonymous

opppss... sure2. thanks for the info :)

- TuringTest

welcome :)
just post separately, I bet across will help

- across

To avoid redundancy, I will just remind you that you are looking for a particular solution\[y_p(x)=-y_1(x)\int\frac{y_2(x)g(x)}{W(y_1,y_2)(x)}dx+y_2(x)\int\frac{y_1(x)g(x)}{W(y_1,y_2)(x)}dx.\]In this case,\[y_1(x)=\cos(x),\]\[y_2(x)=\sin(x),\]\[g(x)=\csc^3(x)\cot(x)\text{, and}\]\[W(y_1,y_2)=\begin{vmatrix}\cos(x)&\sin(x)\\-\sin(x)&\cos(x)\end{vmatrix}=\cos^2(x)+\sin^2(x)=1.\]Therefore,\[y_p(x)=\sin(x)\int\cos(x)\csc^3(x)\cot(x)dx-\cos(x)\int\sin(x)\csc^3(x)\cot(x)\]\[=\frac{1}{2}\cos(x)\cot^2(x)-\frac{1}{3}\cos(x)\cot^2(x)\]\[=\frac{1}{6}\cos(x)\cot^2(x).\]

- across

If it is not right, then I may have made a sign error, but you get the idea.

- anonymous

but the answer is wrong :(

- TuringTest

you said
"y=C1cosx+C2sinx+1/6cotxcscx that's my answer but i dont know where does 1/6cotxcscx came from. :)"
but we are getting
y=C1cosx+C2sinx+1/6cot^2x*cscx
(across and I get the exact same result)
so I am betting on a typo somewhere perhaps

- anonymous

oh maybe they just simplfy it more? but our answer at first is correct? right?

- TuringTest

\[\frac16\cot^2x\csc x\neq\frac16\cot x\csc x\]so it's not a simplification issue

- TuringTest

Since across and I get the exact same result I am going to say that we have provided the correct answer to the posted problem.
Be sure the problem as you posted it is free of typos

- anonymous

hmmm.. what do u mean typos? :)

- TuringTest

are you \(sure\) the original problem is\[y''+y=\csc^3x\cot x\]?
If you wrote it wrong, that would be a "typo"
I think the problem is written correct though, because this makes the integration easy, which means there is likely a typo in the solution in the back of your book; i.e. it should be cot^2x not cotx
(an easy typo for a book to make)

- anonymous

ahh... yep that's the problem. hmm. so which answer should i follow? YOURS? or THE BOOK? HAHA

- across

Just talk to your professor if you seek further clarification.

- anonymous

hmmm.. but this is a bring home exam. i hope he will tell me that the book is wrong HAHA.. i think i will follow your process. and i will consult also my prof tom. thank you for solving it. i hope i can count on you next time :)

- TuringTest

Across and I came to the same answer individually.
I'm self-taught in DE's and make mistakes at times, but across goes to MIT, so I'm pretty sure the book is wrong

- TuringTest

and it looks like wolfram agrees with across and I, so I'm going to say book is wrong: final answer
http://www.wolframalpha.com/input/?i=y%22%2By%3Dcsc%5E3xcotx

- anonymous

HAHAHA. great! THANKS to you two :)

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