2002 I 2

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We have the equation:

\frac{d^{3}y}{dx^{3}}+x^{2}y-\sinh\left(x\right)=0

The homogenous equation is:

\frac{d^{3}y}{dx^{3}}+x^{2}y=0

Taking y˜eS:

S^{\prime\prime\prime}+3S^{\prime}S^{\prime\prime}+\left(S^{\prime}\right)^{3}+x^{2}=0

Taking a balance between \left(S^{\prime}\right)^{3} and x2, \theta=0,\,2\pi/3,\,4\pi/3. Then S=-\frac{3}{5}e^{i\theta}x^{5/3}+g:

\left(\frac{2}{9}e^{i\theta}x^{-4/3}+g^{\prime\prime\prime}\right)+3\left(-e^{i\theta}x^{2/3}+g^{\prime}\right)\left(-\frac{2}{3}e^{i\theta}x^{-1/3}+g^{\prime\prime}\right)+\left(-e^{i\theta}x^{2/3}+g^{\prime}\right)^{3}+x^{2}=0

Expanding:

\frac{2}{9}e^{i\theta}x^{-4/3}+g^{\prime\prime\prime}+2e^{2i\theta}x^{1/3}-3e^{i\theta}x^{2/3}g^{\prime\prime}-2e^{i\theta}x^{-1/3}g^{\prime}+3e^{2i\theta}x^{4/3}g^{\prime}-3e^{i\theta}x^{2/3}\left(g^{\prime}\right)^{2}+\left(g^{\prime}\right)^{3}=0

Balance between 2e2iθx1 / 3 and 3e^{2i\theta}x^{4/3}g^{\prime} gives g=-\frac{2}{3}\ln x. The homogenous solution can then be written:

y_{homog}=Ae^{-3x^{5/3}/5}+\left[B\sin\left(\frac{3\sqrt{3}x^{5/3}}{10}\right)+C\cos\left(\frac{3\sqrt{3}x^{5/3}}{10}\right)\right]e^{3x^{5/3}/10}

The inhomogenous equation is:

\frac{d^{3}y}{dx^{3}}+x^{2}y-\sinh\left(x\right)=0

For large x, the last homogenous terms dominate (since they go like e^{x^{5/3}}), unless B = C = 0. Then we guess a balance between x2y and ex / 2, giving:

y_{part}=\frac{e^{x}}{2x^{2}}

Which does dominate. The general solution is then:

y=Ae^{-3x^{5/3}/5}+\left[B\sin\left(\frac{3\sqrt{3}x^{5/3}}{10}\right)+C\cos\left(\frac{3\sqrt{3}x^{5/3}}{10}\right)\right]e^{3x^{5/3}/10}+\frac{1}{2x^{2}}e^{x}

This page was recovered in October 2009 from the Plasmagicians page on Generals_2002_I_2 dated 20:13, 25 April 2007.

Retrieved from "https://qed.princeton.edu/main/PlasmaWiki/Generals/2002_I_2"
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