求微分方程

的解，满足当 $x=0$ 时 $y=1$。将答案写成 $y=f(x)$ 的形式。

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。

先乘积分因子 $e^{3x}$。右边用两次分部积分，第二次会出现原来的积分，移项即可求出。

The equation is already in standard linear form, so an integrating factor is $e^{\int3\,\mathrm{d}x}=e^{3x}$. Multiplying by $e^{3x}$ gives $(ye^{3x})'=e^{3x}\sin x$. Write $I=\int e^{3x}\sin x\,\mathrm{d}x$ and $J=\int e^{3x}\cos x\,\mathrm{d}x$, taking constants at the end. For $I$, take $u=e^{3x}$ and $\mathrm{d}v=\sin x\,\mathrm{d}x$, so $v=-\cos x$. For $J$, take $u=e^{3x}$ and $\mathrm{d}v=\cos x\,\mathrm{d}x$, so $v=\sin x$. Integration by parts gives

Substituting $J$ into the first equation gives $I=-e^{3x}\cos x+3e^{3x}\sin x-9I$. Moving $-9I$ to the left yields $10I=e^{3x}(3\sin x-\cos x)$. Hence

At $x=0$, $y=1$, so $C=\frac{11}{10}$. Therefore

也可把 $\sin x$ 看成 $e^{\mathrm{i}x}$ 的虚部。乘积分因子后，先积分一个复指数，取虚部时答案就回到实数。

Using the same integrating factor $e^{3x}$, we have

Setting this equal to $ye^{3x}$ and imposing $y(0)=1$ gives $C=11/10$ and $y=(3\sin x-\cos x+11e^{-3x})/10$.