求微分方程

的解，使得当 $x = 1$ 时 $y = \frac{1}{2}\pi$。将答案写成 $y = f(x)$ 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。

在含初始点的 $x>0$ 区间内，除以 $x$ 化成线性方程。积分因子为 $1/x$，乘上后对右边做分部积分。

For $x>0$, the equation is $y'-y/x=2x\arctan x$. The integrating factor is

Multiplying by it gives

For $\int2\arctan x\,\mathrm{d}x$, take $u=\arctan x$ and $\mathrm{d}v=2\,\mathrm{d}x$, so $v=2x$ and $\mathrm{d}u=\mathrm{d}x/(1+x^2)$. Integration by parts gives

At $x=1$, we have $\frac{\pi}{2}=\frac{\pi}{2}-\ln2+C$, so $C=\ln2$. Therefore