標籤彙整： Intro to DE

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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微分方程

【微分方程導論】筆記10

2019-11-27 Sin-iu 發表迴響

Let’s look at some examples to illustrate the s-axis translation theorem (Theorem 3 introduced in the last lecture).

Example 1

Find the inverse Laplace transform of $\displaystyle \frac{1}{s^2-4s+5}$ .

Solution
Let $\displaystyle F(s) = \frac{1}{s^2+1}$ and $\displaystyle f(t) = \mathcal{L}^{-1}\{F(s)\} = \sin t$ . Then because

$\displaystyle \mathcal{L}^{-1}\left\{\frac{1}{s^2-4s+5}\right\} = \mathcal{L}^{-1}\left\{\frac{1}{(s-2)^2+1}\right\} = \mathcal{L}^{-1}\{F(s-2)\}$ ,

by s-axis translation theorem, $\displaystyle \mathcal{L}^{-1}\{F(s-2)\} = e^{2t}f(t) = e^{2t}\sin t.$ ❏

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微分方程

【微分方程導論】筆記9

2019-11-21 Sin-iu 發表迴響

Theorem 1

(the Laplace transform of the first derivative of $\textit{\textbf{f}}$ )
Suppose:
(1) $f$ is continuous on some interval and of exponential order $\alpha$ ; and
(2) $f'$ is piecewise continuous on some interval.
Then $\mathcal{L}\{f'(t)\}$ exists for $s > \alpha$ and $\displaystyle \mathcal{L}{f'(t)} = s\mathcal{L}{f(t)} - f(0).$

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