Math Is Fun Forum

numen

Member

Registered: 2006-05-03

Posts: 115

Solving this DE (trigonometry)

the problem:
y' cos(x) + 2y sin(x) = ([sin(x)]^3)/(cos(x))

I know I could divide by cos(x) and calculate an integration factor, which I think would be 1/[cos(x)]^2 ?

Then I get: D(y/([cos(x)]^2)) = ([sin(x)]^3)/([cos(x)]^3)

Thus, the integral follows: ∫ ([sin(x)]^3)/([cos(x)]^4) dx, which I'm not sure how to solve...

Any ideas? I'm kinda stuck with this one, might be something wrong somewhere. My head just keeps spinning around

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Bang postponed. Not big enough. Reboot.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Solving this DE (trigonometry)

http://www.mathsisfun.com/forum/viewtop … 635#p32635

Using that formula, I get:

This ugly equation comes out to be:

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

numen

Member

Registered: 2006-05-03

Posts: 115

Re: Solving this DE (trigonometry)

I've learned to not use formulas straight off like that, I learn nothing from it. Kinda ugly, yeah. But thanks anyway for showing that formula, might get in handy, though I preferably do everything from scratch.

Do you know, though, how to continue from where I left? Anyone else? I'll think about it some more, I simply can't sleep until I've figured it out =P

btw, is that LaTeX? How do I put it in here?

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Bang postponed. Not big enough. Reboot.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Solving this DE (trigonometry)

Since the equation is not seperable and not homogenous, I don't believe there is any other way to solve it.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,385

Re: Solving this DE (trigonometry)

numen.
Here is a link on how to use  LaTeX, in this forum or any other place.

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Solving this DE (trigonometry)

I've written the proof or the derivation in post 3, following Ricky's fomula, so you can check my procedure and simply duplicate it before showing the formula. And that will make your solution complete.

Last edited by George,Y (2006-05-04 15:52:44)

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X'(y-Xβ)=0

numen

Member

Registered: 2006-05-03

Posts: 115

Re: Solving this DE (trigonometry)

It's solved now. Thanks George, even though it's not what I was looking for... again.

Anyway, the (easier) solution is y = 1/3cos(x)-cos(x)+C[cos(x)]^2. You get it by rewriting the integral I ended up with by using trigonometry rules and then use variable substitution to simplify the process.

Now I can go to sleep again :] thanks ganesh for that link, will have a look.

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Bang postponed. Not big enough. Reboot.

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Solving this DE (trigonometry)

Hey, I know how to solve it now, too!
The keys are
a) the derivative of cos²(x) is -2cos(x)sin(x)
b)

Your question is so inspiring!  I used to think this kinda trig and derivative mix is my last-to-solve.

Last edited by George,Y (2006-05-05 14:04:53)

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X'(y-Xβ)=0

yttrium88

Member

Registered: 2005-12-01

Posts: 20

Re: Solving this DE (trigonometry)

As far as that first integral, i would just replace (sin x)^3 with (1-(cos x)^2)(sin x) and sub in with  u=cos x
du=-sinx dx