(a) 证明

(b) 求微分方程

的特解，已知当 $x=0$ 时，$y=1$ 且 $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{4}{3}$。

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。

代入双曲正弦和余弦的指数定义。相加后，含 $e^{-x}$ 的项抵消。

The exponential definitions give

Taking the positive square root gives $(\cosh x+\sinh x)^{1/2}=e^{x/2}$.

用 (a) 把右边化成指数。先由辅助方程求互补函数，再试 $ke^{x/2}$ 求特解，最后使用两个初始条件。

The auxiliary equation $m^2+m+3=0$ has roots $-\frac12\pm\frac{\sqrt{11}}2\mathrm{i}$. With $\omega=\frac{\sqrt{11}}2$, the complementary function is $e^{-x/2}(A\cos\omega x+B\sin\omega x)$. For the particular integral $y_p=ke^{x/2}$, we have $y_p'=\frac{k}{2}e^{x/2}$ and $y_p''=\frac{k}{4}e^{x/2}$. Substituting gives

The general solution and its derivative are

The initial conditions give

The first equation gives $A=-\frac13$. The second then gives $\omega B=\frac12$, so $B=1/\sqrt{11}$. Therefore