## How To

### How to get solution by Method of Variation of Parameters?

by » 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?

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Joined: Tue Feb 07, 2012 7:12 am

### Re: How to get solution by Method of Variation of Parameters?

by » Tue Feb 07, 2012 6:36 pm

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?

by » 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)

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Joined: Tue Feb 07, 2012 7:12 am

### Re: How to get solution by Method of Variation of Parameters?

by » 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.
WolframAlphaMathHelp

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### Re: How to get solution by Method of Variation of Parameters?

by » 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?

Posts: 4
Joined: Tue Feb 07, 2012 7:12 am

### Re: How to get solution by Method of Variation of Parameters?

by » Tue Feb 21, 2012 4:51 pm

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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