变量 $x$ 和 $y$ 满足微分方程

已知当 $x=0$ 时，$y=0$。

解此微分方程，并求当 $x=2$ 时 $y$ 的值。

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。

分离变量后，把右边拆成对数型与反正切型积分。初值确定常数，再指数化求 $y$。

Separating variables,

Integrating,

Here the logarithmic term comes from $\int x/(x^2+3)\,\mathrm{d}x$, and the inverse tangent term from $-2\int1/(x^2+3)\,\mathrm{d}x$.

Using $x=0$, $y=0$,

Multiplying the particular solution by $-3$ and combining logarithms,

Writing

the solution is

This solution applies on the interval containing $x=0$ where the outer logarithmic argument is positive. It remains positive up to $x=2$: its derivative is $-3(x-2)/(x^2+3)\geqslant0$ on $[0,2]$, and its initial value is $1$.

At $x=2$,

Hence