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Topic: differential equation (Read 2045 times) |
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hparty
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differential equation
« on: Sep 21st, 2010, 5:06pm » |
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Consider the second order differential equations
(1) y''(x) + y(x) +y^3(x) = 0 , x \in R
(2) y''(x) + y'(x)+y(x) +y^3(x) = 0, x\in R
Prove that (1) has a solution for all x \in R.
What about the equation in (2).
Thanks for help in advance
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hparty
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Re: differential equation
« Reply #1 on: Sep 23rd, 2010, 5:34pm » |
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Any hint?? 
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Aryabhatta
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Re: differential equation
« Reply #2 on: Sep 23rd, 2010, 11:11pm » |
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y(x) = 0 is a trivial solution to both.
To get a non-trivial solution to 1) the below might help.
Multiply by 2y'(x).
You get
d( (y')^2)/dx = -2y'(y + y^3)
Now integrate, you get
(y')^2 = -y^2 - y^4/2 + C
Thus
y' = sqrt(C - y^2 -y^4/2)
i.e.
y'/(sqrt(C-y^2 -y^4/2) = 1
Integrate again (I have no clue how to do that).
Perhaps it will go somewhere.
Didn't think about 2).
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towr
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Re: differential equation
« Reply #3 on: Sep 24th, 2010, 4:03am » |
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Perhaps there is a non-constructive proof? I.e. without having to find the solution. (Although wolframalpha does give one, but it involves elliptic curves.)
For example, you can interpret the first equation as describing a physical system F(t) = - [s(t) + s(t)^3]. I don't think there's a reason why such a system couldn't exist, with an opposing force growing as a third degree polynomial of the distance from equilibrium position.
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Wikipedia, Google, Mathworld, Integer sequence DB
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Aryabhatta
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Re: differential equation
« Reply #4 on: Sep 24th, 2010, 10:27am » |
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on Sep 24th, 2010, 4:03am, towr wrote:Perhaps there is a non-constructive proof? I.e. without having to find the solution. (Although wolframalpha does give one, but it involves elliptic curves.)
For example, you can interpret the first equation as describing a physical system F(t) = - [s(t) + s(t)^3]. I don't think there's a reason why such a system couldn't exist, with an opposing force growing as a third degree polynomial of the distance from equilibrium position. |
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Yup, I am pretty sure there will be some existence theorem which will imply the existence of a non-trivial solution. No idea what though 
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