(a) 求微分方程

的通解。

(b) 证明对于较大的正 $t$ 值以及任意初始条件，

其中常数 $R$ 和 $\varphi$ 待确定。

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 $-5$，所以互补函数要包含 $te^{-5t}$。特解试作 $p\sin t+q\cos t$，比较两个系数。

The auxiliary equation is $(m+5)^2=0$, so the complementary function is $(At+B)e^{-5t}$. For a particular integral, try $x=p\sin t+q\cos t$. Then $x'=p\cos t-q\sin t$ and $x''=-p\sin t-q\cos t$. Substitution gives

The second equation gives $q=-5p/12$. Substituting into the first gives

so $p=12$ and $q=-5$. Thus

含初始条件的指数项随 $t$ 增大趋于零。对留下的正弦和余弦比较系数，求出振幅和相位。

For any fixed $A,B$, $(At+B)e^{-5t}\to0$ as $t\to\infty$. Matching $12\sin t-5\cos t$ with $R\sin(t-\varphi)$ gives

Taking $R>0$ gives $R=\sqrt{12^2+5^2}=13$. Both $\sin\varphi$ and $\cos\varphi$ are positive, so we choose the first-quadrant phase $\varphi=\arctan(5/12)\approx0.395$ radians. The approximation is