求微分方程

的特解，已知当 $x=0$ 时，$y=-4$ 且 $\frac{\mathrm{d}y}{\mathrm{d}x}=3$。

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。

辅助方程有重根 $1$，所以互补函数包含 $xe^x$。特解同时试正弦和余弦，再用两个初始条件求常数。

The auxiliary equation $(m-1)^2=0$ has a repeated root, so the complementary function is $(A+Bx)e^x$. For a particular integral, take $y=p\sin x+q\cos x$. Then $y_p'=p\cos x-q\sin x$ and $y_p''=-p\sin x-q\cos x$. Since $y_p''+y_p=0$ for this trial, the left-hand side reduces to

Matching $4\cos x$ gives $p=-2$ and $q=0$. The general solution and its derivative are

The initial conditions give $A=-4$ and $A+B-2=3$. Therefore