变量 $x$ 和 $\theta$ 满足微分方程

已知当 $\theta=0$ 时，$x=1$。

解此微分方程，得到用 $\theta$ 表示 $x^2$ 的表达式。

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。

把含 $x$ 与含 $\theta$ 的项分到两侧。积分后利用初始值定常数，再指数化去掉对数，写成 $x^2$ 的形式。

Separating variables gives

Integrating,

The separated equation is obtained where the original fractions are defined. Its integrated relation is continuous at $\theta=0$, so imposing the continuous initial value $x(0)=1$ gives

Hence

Exponentiating,

Therefore

The branch through the initial point has $x>0$ and $-\pi/2<\theta<\pi/2$. It satisfies the original differential equation for $\theta\ne0$ in this interval and extends smoothly to the initial point; the original quotient involving $\tan\theta$ is not evaluated at $\theta=0$.