(a) 将 $y=\sec^3\theta$ 写为 $\frac1{\cos^3\theta}$，证明 $\frac{\mathrm{d}y}{\mathrm{d}\theta}=3\sin\theta\sec^4\theta$。

(b) 变量 $x$ 和 $\theta$ 满足微分方程

已知当 $\theta=\frac13\pi$ 时，$x=3$。

解此微分方程，求 $x=0$ 时 $\cos\theta$ 的值。答案保留 $3$ 位有效数字。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

按题目要求先写成余弦的负三次方，再用链式法则展示两个负号相消。

Writing $y=1/\cos^3\theta=(\cos\theta)^{-3}$,

分离变量后，左边用(a)积分。右边拆成对数型和反正切型，代入初值后只需在目标点求正割三次方。

Separating variables,

Using (a) on the left, split the right-hand side as

The first term integrates to $\frac12\ln(x^2+9)$ because the denominator has derivative $2x$. The second integrates to $\tan^{-1}(x/3)$, whose derivative is $3/(x^2+9)$. Thus

At $x=3$, $\theta=\pi/3$, so

Thus

At $x=0$,

This quantity is positive. Taking the real cube root gives

Therefore $\boxed{\cos\theta=0.601}$ to 3 significant figures.

也可在分离变量后把 $x$ 换成 $3\tan\phi$，把有理积分变成正切与常数之和。

As above, separation and (a) give

Let $x=3\tan\phi$ with $0\leqslant\phi\leqslant\pi/4$ as $x$ ranges from $0$ to $3$, so $\mathrm{d}x=3\sec^2\phi\,\mathrm{d}\phi$ and $\cos\phi>0$. Then

Hence

The initial value $x=3$ gives $\phi=\pi/4$ and $\theta=\pi/3$. Therefore

At $x=0$, $\phi=0$, so

Thus