求微分方程

满足当 $x=0$ 时 $y=3$ 的特解。将答案写成 $y=\mathrm{f}(x)$ 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。

乘积分因子 $e^x$，把左边写成乘积的导数。积分后代入初值。

The equation is in standard linear form, with coefficient $1$ for $y$. An integrating factor is $e^{\int1\,\mathrm{d}x}=e^x$. Multiplying by $e^x$ gives $(ye^x)'=e^{3x/2}$. Integrating the right-hand side gives $\int e^{3x/2}\,\mathrm{d}x=\frac23e^{3x/2}+C$, so $ye^x=\frac23e^{3x/2}+C$. Since $y(0)=3$, we have $C=3-\frac23=\frac73$, so

也可先解齐次方程，再试与右端同形的特解 $ke^{x/2}$。相加后用初值定常数。

The homogeneous equation $y'+y=0$ has solution $Be^{-x}$. Substituting $ke^{x/2}$ into the original equation gives $(k/2+k)e^{x/2}=e^{x/2}$, so $k=\frac23$. Thus $y=Be^{-x}+\frac23e^{x/2}$. At $x=0$, $B+\frac23=3$, yielding $B=\frac73$.