求微分方程

的特解，已知当 $x=0$ 时，$y=2$ 且 $\frac{\mathrm{d}y}{\mathrm{d}x}=-8$。

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。用二次多项式试一个特解，再用初始条件确定互补函数中的常数。

The auxiliary equation $m^2+2m+3=0$ gives $m=-1\pm\sqrt2\,\mathrm{i}$. The complementary function is $e^{-x}(A\cos(\sqrt2x)+B\sin(\sqrt2x))$. For $y_p=px^2+qx+r$, we have $y_p'=2px+q$ and $y_p''=2p$. Substituting gives

Thus $p=9$, $q=-12$ and $r=2$. The general solution and its derivative are

The initial conditions give $A+2=2$ and $-A+\sqrt2 B-12=-8$, hence $A=0$ and $B=2\sqrt2$. Therefore