求微分方程

的特解，已知当 $x = 0$ 时，$y = 1$ 且 $\frac{\mathrm{d}y}{\mathrm{d}x} = 0$。

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。

解辅助方程得到衰减振荡的互补函数。右边为 $13e^{3x}$，特解试作 $ke^{3x}$，最后用两个初始条件。

The auxiliary equation $m^2+4m+5=0$ gives $m=-2\pm\mathrm{i}$. The complementary function is $e^{-2x}(A\cos x+B\sin x)$. For a particular integral $y_p=ke^{3x}$, the derivatives are $y_p'=3ke^{3x}$ and $y_p''=9ke^{3x}$. Substitution and division by $e^{3x}$ give $(9+12+5)k=13$, so the particular integral is $\frac12e^{3x}$.

The general solution and its derivative are

Now $y(0)=1$ gives $A+\frac12=1$. The derivative at $0$ gives $B-2A+\frac32=0$, so $A=\frac12$ and $B=-\frac12$. Therefore