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February 14th, 2012, 07:02 AM   #1
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Solve the following ODEs...

1. (5y-(5/2)xy^3) dx -dy =0

2. (-3x+y+6)dx + (x+y+dy) = 0

Yea I got these totally wrong.. can I get some pointers?
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February 14th, 2012, 09:18 AM   #2
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Re: Solve

Hello, FreaKariDunk!

There is a typo in #2.






The equation is Reducible to Homogenous . . . a rather tedious procedure.








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February 14th, 2012, 09:58 AM   #3
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Re: Solve

It is supposed to be (-3x + y + 6)dx - (x+y+2)dy = 0

Sorry.
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February 14th, 2012, 07:22 PM   #4
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Re: Solve

1.)

We can express this in the form:



and we now have a Bernoulli equation with n = 3, and . To transform this ODE into a linear equation, we first divide by to obtain:



Next, we make the substitution . Since , the transformed equation is:

or



Now we have a linear equation, for which the integrating factor is :





Integrating with respect to x yields:







Back substituting for v, we have the implicit relation:



Not included in the last equation is the solution that was lost in the process of dividing by above.

2.) I am assuming you mean this based on your initial post.

This is an equation with linear coefficients and we will use the translation of axes where h and k satisfy the system:





Solving this system, we find h = 1 and k = -3. Hence, we let:



.

Since and substitution into the ODE for x and y yields

or

The above equation is homogeneous, so we let , and so:

and substituting for we obtain:



Separating variables gives:





from which it follows that:



When we substitute back in for z, u, and v, we find:











This last equation gives an implicit solution to the original ODE.
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February 15th, 2012, 07:55 AM   #5
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Re: Solve the following ODEs...

Thanks for your help Mark.

Btw, you're always picking on me. :P I couldn't find my thread because you changed the name. Thanks for clarifying! You're my favorite mod... along with the others.
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February 15th, 2012, 12:29 PM   #6
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Re: Solve the following ODEs...

Awww...I didn't mean to pick on you!!

I simply felt that this topic deserved a more descriptive title than "Solve."
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February 15th, 2012, 07:20 PM   #7
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Re: Solve the following ODEs...

Solve worked because I needed a quick subject so I could have valentines lunch with my White Trash Wife. It's hard trying to be as awesome as you.
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February 15th, 2012, 08:28 PM   #8
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Re: Solve the following ODEs...

What was awesome was finding both of these problems as worked out examples in my old DiffEq textbook!
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February 17th, 2012, 02:22 PM   #9
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The guessed equation can simply be integrated to give -(3/2)x + 6x + xy + (1/2)y = c.
The corrected equation can simply be integrated to give -(3/2)x + 6x + xy + (1/2)y + 2y = c.
The LHS of the second of the above solutions can be factorized, giving (3x + y)(-x + y + 4)/2 = c.
When c = 0, this gives (3x + y)(-x + y + 4)/2 = 0, but this should be converted to the solutions y = -3x and y = x - 4.
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February 17th, 2012, 03:52 PM   #10
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Re: Solve the following ODEs...

Quote:
Originally Posted by MarkFL
What was awesome was finding both of these problems as worked out examples in my old DiffEq textbook!

So uh... lemme borrow that pleaseee? I would love to find the solutions manual to the text book he gave us online.
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