Quod Erat Demonstrandum

2014/06/11

M2 某題

Filed under: Additional / Applied Mathematics,NSS — johnmayhk @ 9:06 上午
Tags: ,

M2 學生問以下一題

The slope at any point (x,y) of a curve is given by

\frac{dy}{dx}=y(2x+9).

If the curve lies above the x-axis, and it passes (0,8), find the equation of the curve.

對於以前有修應數的中學生,解這個一階線性常微分方程是最初級的知識,利用分離法便可:

\frac{dy}{y}=(2x+9)dx \Rightarrow \ln y=\int (2x+9)dx (y > 0

但對現在的學生,那是難題。

但如果給一個提示,比如

\frac{d(\ln y)}{dx}=?

現在的學生也輕易處理如下:

\frac{d(\ln y)}{dx}=\frac{1}{y}\frac{dy}{dx}

\frac{dy}{dx}=y(2x+9)

變成

y\frac{d(\ln y)}{dx}=y(2x+9)

\frac{d(\ln y)}{dx}=2x+9

\ln y=\int (2x+9)dx

\dots

johnmayhk-mr-bean
祝大家考試:盡力而為,問心無愧

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