Math Is Fun Forum

freddogtgj

Member

Registered: 2006-12-02

Posts: 54

Differential Equation

I have this particular question which i'm having trouble with... could someone help me?

"Solve the following initial value problem
(1+x)y' + y^2 - 1 = 0,        y(0)=0"

Thanks in advance!

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Differential Equation

This differential equation is of the variables-separable type. You can separate the variables and integrate.

freddogtgj

Member

Registered: 2006-12-02

Posts: 54

Re: Differential Equation

If you don't mind checking my answer:

arctanh(y) = ln (x+1)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Differential Equation

That’s what I’ve got as well.

Since

you can also write the answer as

Last edited by JaneFairfax (2007-03-10 12:01:05)

freddogtgj

Member

Registered: 2006-12-02

Posts: 54

Re: Differential Equation

Cheers!

freddogtgj

Member

Registered: 2006-12-02

Posts: 54

Re: Differential Equation

I have another similar question this time, but it's proving too be a little bit more difficult:

y' + tan(x)y + x = 0,        y(0)=2

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Differential Equation

This is an equation of the form

I believe there’s a standard method for solving it. (You know, the one involving particular integrals and complementary functions?)

EDIT: Er, I found the formula to use:
http://eqworld.ipmnet.ru/en/solutions/ode/ode0103.pdf

Last edited by JaneFairfax (2007-03-11 04:14:14)

freddogtgj

Member

Registered: 2006-12-02

Posts: 54

Re: Differential Equation

I'm not sure if that applies to this question

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Differential Equation

Why not? The solution is

where C is a constant to be determined. You just have to find a way to integrate xsecx – so good luck with that one.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Differential Equation

I have a very strong suspicion that you got one of the signs in your equation wrong. Instead of

I’m very sure it should be

with a minus sign for the tanx. Then instead of xsecx you would be integrating xcosx – which is really a piece of cake. The solution to the second equation above satisfying the given initial condition would then be

Frankly I don’t see how you can integrate xsecx without making so much of a mess that the question becomes not worth answering at all.

Last edited by JaneFairfax (2007-03-11 05:43:13)

freddogtgj

Member

Registered: 2006-12-02

Posts: 54

Re: Differential Equation

I agree with you as well but the question seems to be what i have typed out

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Differential Equation

Well, then you’ll have to find a way to integrate xsecx. If you integrate by parts, you’ll have to integrate the integral of secx, which is ln|secx+tanx|. I’m not sure how you do it, really.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Differential Equation

Mathematica gives:

Therefore, you probably wrote the question wrong.  QED.

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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..."