(a) 用代换 $u=\cosh x$，求 $\displaystyle\int\sinh^3x\,\mathrm{d}x$。

(b) 求微分方程

满足当 $x=\ln\frac{1}{3}$ 时 $y=0$ 的特解。将答案写成 $y=\mathrm{f}(x)$ 的形式。

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。

将一个 $\sinh x$ 和 $\mathrm{d}x$ 合并成 $\mathrm{d}u$，再用 $\sinh^2x=\cosh^2x-1$ 替换其余因子。

With $u=\cosh x$, $\mathrm{d}u=\sinh x\,\mathrm{d}x$. Therefore

初值点在 $x<0$。这里 $\sinh x$ 非零，而且它的导数除以它本身就是 $\coth x$，可直接用作积分因子。

The equation is already in standard linear form. On $x<0$, $\int\coth x\,\mathrm{d}x=\ln|\sinh x|+C$. An integrating factor is $e^{\int\coth x\,\mathrm{d}x}=|\sinh x|$, or equivalently its constant multiple $\sinh x$. Using the latter gives

Integrating and using part (a),

At $x=\ln(1/3)$, $\cosh x=\frac12(1/3+3)=\frac53$ and $y=0$. Thus $0=\frac13(\frac53)^3-\frac53+C$, so $C=\frac{10}{81}$. Therefore