求微分方程

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

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。

先解辅助方程得到互补函数。用二次多项式试特解，最后利用两个零初值确定常数。

The auxiliary equation $2m^2+2m+1=0$ gives $m=-\frac12\pm\frac12\mathrm{i}$. The complementary function is

For the particular integral, take $y_{\mathrm p}=px^2+qx+r$. Then $y_{\mathrm p}'=2px+q$ and $y_{\mathrm p}''=2p$, giving

Matching coefficients gives $p=4$, $4p+q=3$ and $4p+2q+r=3$, so $q=-13$ and $r=13$. The general solution and its derivative are

At $x=0$, the two initial conditions give

Hence $A=-13$ and $B=13$, and the solution is