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。

辅助方程有一对共轭复根。用二次多项式试特解，再用 $y(0)$ 和 $y'(0)$ 解出互补函数的两个常数。

The auxiliary equation $3m^2+2m+1=0$ gives $m=-\frac13\pm\frac{\sqrt2}{3}\mathrm{i}$. The complementary function is

For a particular integral $y_p=px^2+qx+r$, we have $y_p'=2px+q$ and $y_p''=2p$. Substituting and comparing coefficients gives $p=1$, $4p+q=0$ and $6p+2q+r=0$. Hence the particular integral is $x^2-4x+2$. Write $w=\sqrt2/3$. The general solution and its derivative are

From the initial conditions,

The first equation gives $A=-2$. Then $2/3+\sqrt2 B/3-4=0$, so $\sqrt2 B=10$ and $B=5\sqrt2$. The solution is