Generals 2005 I 1

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We consider the differential equation:

$\frac{d^{2}y}{dx^{2}}+\frac{1}{x^{2}}\frac{dy}{dx}+\frac{1}{x^{2}}y=0$

[edit] Part a

For $x ˜0$ , the equation is singular, so we guess $y ˜ e S$ :

$\left(S^{\prime\prime}+\left(S^{\prime}\right)^{2}\right)+\frac{S^{\prime}}{x^{2}}+\frac{1}{x^{2}}=0$

Balance between $\left(S^{\prime}\right)^{2}$ and $S^{\prime}/x^{2}$ gives $S^{\prime}=-x^{-2}$ , which is dominant. There is an alternate balance between $S^{\prime}/x^{2}$ and $x - 2$ , giving $S^{\prime}=-1$ . Finding the next term $g$ for the first case:

$\left(\frac{2}{x^{3}}+g^{\prime\prime}-\frac{2g^{\prime}}{x^{2}}+\left(g^{\prime}\right)^{2}\right)+\frac{g^{\prime}}{x^{2}}+\frac{1}{x^{2}}=0$

So balance between $2 / x 3$ and $g^{\prime}/x^{2}$ gives $g^{\prime}=x^{-1}$ . The other term is already less singular that $ln x$ . The leading asymptotic forms are then:

$y\sim xe^{1/x},\quad e^{-x}$

[edit] Part b

For $x\rightarrow\infty$ , the equation is again singular, so, guessing $y\sim e^{S}</math>:

$\left(S^{\prime\prime}+\left(S^{\prime}\right)^{2}\right)+\frac{S^{\prime}}{x^{2}}+\frac{1}{x^{2}}=0$

This time there is balance between $S^{\prime\prime}+\left(S^{\prime}\right)^{2}+x^{-2}$ , giving $S\sim\frac{1}{2}\left(-1\pm i\sqrt{3}\right)\ln x$ . So:

$y\sim x^{-\left(1\pm i\sqrt{3}\right)/2}$

For $x\rightarrow\infty$ the equation is not singluar, so, writing $y=\sum_{n}a_{n}x^{-n-\alpha}</math>:

$\sum_{n}a_{n}\left(n+\alpha\right)\left(n+\alpha+1\right)x^{-n-2-\alpha}-\sum_{n}a_{n}\left(n+\alpha\right)x^{-n-3-\alpha}+\sum_{n}a_{n}x^{-n-2-\alpha}=0$

Shifting: