Intro to DE / Lecture Note 12

2019-12-11 Sin-iu 發表迴響

Consider a second order linear homogeneous differential equation

$\displaystyle a_2(x)y''+a_1(x)y'+a_0(x)y = 0.$

As defined in the last lecture, if $\displaystyle \frac{a_1(x)}{a_2(x)}$ and $\displaystyle \frac{a_0(x)}{a_2(x)}$ are analytic at $x_0$ , then $x_0$ is an ordinary point; otherwise, $x_0$ is an singular point (singularity).

Theorem 1

If $x_0$ is an ordinary point of Equation (1), then there exist two linearly independent solutions of the form $\displaystyle \sum_{n=0}^\infty c_n(x-x_0)^n$ , where the series converges on $|x-x_0| < R$ for some $R>0$ .

Note
For convenience, it’s often to let $x_0 = 0$ . When $x_0 = 0$ is not an ordinary point, then make a shift!

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2019-12-05 Sin-iu 發表迴響

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留學

動機信寫作教學彙整

2019-12-01 Sin-iu 發表迴響

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