求微分方程

的特解，已知当 $x = 0$ 时，$y = \frac{\mathrm{d}y}{\mathrm{d}x} = 0$。

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。

由辅助方程求互补函数，用二次多项式试出一个特解。相加后用 $y(0)$ 和 $y'(0)$ 确定两个常数。

The auxiliary equation $5m^2+2m+1=0$ has roots $-\frac15\pm\frac25\mathrm{i}$. The complementary function is

For a particular integral, substitute $y=px^2+qx+r$, with $y'=2px+q$ and $y''=2p$. Equating coefficients gives

so $p=q=1$ and $r=-9$. The general solution is

Differentiating,

Applying $y(0)=y'(0)=0$ gives

Thus $A=9$, $B=2$ and