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How to get solution by Method of Variation of Parameters?

How to get solution by Method of Variation of Parameters?

Postby aireyriadbonnet » Tue Feb 07, 2012 7:52 am
Hello.i have tried to solve -y"+\pi^2 y = 2\pi^2 sin(\pi x)...and i have gotten the exact solution is sin (\pi x).Very helpful.But i wanna get the step by step answer with Method of Variation of Parameters.Because i don't know about how Wolfram get that,

So i give the input -e^{-\pi x} \int \frac {e^{\pi x}2 \pi^2 sin(\pi x )}{2\pi}dx + e^{\pi x} \int \frac {e^{- \pi x}2 \pi^2 sin(\pi x )}{2\pi}dx.

But in Fact it gives the wrong answer -sin(\pi x) .Would you mind explaining that?
aireyriadbonnet
 
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Re: How to get solution by Method of Variation of Parameters?

Postby WolframAlphaMathHelp » Tue Feb 07, 2012 6:36 pm
aireyriadbonnet,

When I tried your second query, after cleaning it up a bit, and not using Tex notation, I came up with this:

-e^(-pi x) * integral( ( e^(pi x) 2 pi^2 sin(pi x))/(2pi) dx) + e^(pi x) * integral((e^(-pi x) 2 pi^2 sin(pi x))/(2pi) dx)

http://www.wolframalpha.com/input/?i=-e ... %29+dx%29+

Hope that helps.
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Re: How to get solution by Method of Variation of Parameters?

Postby aireyriadbonnet » Thu Feb 16, 2012 3:24 am
sorry but i think thats not the point.what you have done is same with me.you give the wrong answer because the exact solution is y=sin(pi x)
aireyriadbonnet
 
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Re: How to get solution by Method of Variation of Parameters?

Postby WolframAlphaMathHelp » Thu Feb 16, 2012 4:24 pm
Ok. It is not completely obvious what you are trying to do with

-e^(-pi x) * integral( ( e^(pi x) 2 pi^2 sin(pi x))/(2pi) dx) + e^(pi x) * integral((e^(-pi x) 2 pi^2 sin(pi x))/(2pi) dx)

This leads to -sin(pi x).

The particular solution of the ODE that you gave is sin(pi x) and a derivation of this can be seen by clicking on the Show steps button in the Differential equation solutions.

Please let us know if we can be of any more help.
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Re: How to get solution by Method of Variation of Parameters?

Postby aireyriadbonnet » Fri Feb 17, 2012 11:36 pm
by cliking ,the step use undetermined coefisient method,but i wanna get step by variations of parameters.would you please tell me how to get that?
aireyriadbonnet
 
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Re: How to get solution by Method of Variation of Parameters?

Postby WolframAlphaMathHelp » Tue Feb 21, 2012 4:51 pm
aireyriadbonnet,

Sorry, I misunderstood. To solve by variation of parameters, first write the equation in the standard form with leading coefficient 1 in front of the highest derivative:

y"-pi^2 y = -2pi^2 sin(\pi x)

Then solve the homogeneous equation

http://www.wolframalpha.com/input/?i=y% ... E2+y+%3D+0

to get

y(x) = c_1 e^(pi x)+c_2 e^(-pi x)

Next, calculate the Wronskian of the solutions:

http://www.wolframalpha.com/input/?i=+w ... 7D%2C+x%29

which gives -2pi.

Finally, use the general formulas for calculating the inhomogeneous solution:
(eq 50 of http://mathworld.wolfram.com/Second-Ord ... ation.html)

- e^(pi x)* integrate(e^(-pi x) (-2pi^2 sin(pi x))/(-2pi) dx) +
e^(-pi x)* integrate(e^(pi x) (-2pi^2 sin(pi x))/(-2pi) dx)

(this is different from your equation by a couple -1 factors)

http://www.wolframalpha.com/input/?i=-+e%5E(pi+x)*+integrate(e%5E(-pi+x)+(-2pi%5E2+sin(pi+x))%2F(-2pi)+dx)+%2B+++e%5E(-pi+x)*+integrate(e%5E(pi+x)+(-2pi%5E2+sin(pi+x))%2F(-2pi)+dx)

to get sin(x), the same result WA returned immediately for the ODE.
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