三、解答题 (本题满分 15 分, 每小题 5 分)
(1) 已知 $f(x)=\mathrm{e}^{x^{2}}, f[\varphi(x)]=1-x$ 且 $\varphi(x) \geqslant 0$, 求 $\varphi(x)$ 并写出它的定义域.
$$
f[\varphi(x)]=1-x=e^{[\varphi(x)]^{2}} \Rightarrow
$$
构造式子消去上面的 $e$:
$$
\varphi^{2}(x)=\ln (1-x) \Rightarrow \varphi(x)=\sqrt{\ln (1-x)} \Rightarrow
$$
$$
\left\{\begin{array} { l }
{ 1 – x > 0 } \\
{ \operatorname { l n } ( 1 – x ) \geqslant 0 }
\end{array}\right. \Rightarrow \left\{\begin{array} { l }
{ x < 1 } \\
{ 1 – x \geqslant 1 }
\end{array}\right. \Rightarrow
$$
$$
\left\{\begin{array}{l}
x<1 \\
x \leqslant 0
\end{array} \Rightarrow\right.
$$
$$
\varphi(x)=\sqrt{\ln (1-x)}, \quad \mathrm{~ d} x \leqslant 0
$$
(2) 已知 $y=1+x \mathrm{e}^{x y}$, 求 $\left.y^{\prime}\right|_{x=0}$ 及 $\left.y^{\prime \prime}\right|_{x=0}$.
$$
y^{\prime}=e^{x y}+x y \cdot e^{x y} \Rightarrow
$$
$$
x=0 \Rightarrow y=1+0=1 \Rightarrow
$$
$$
x=0, y=1 \Rightarrow y^{\prime}=e^{0}+0=1
$$
$$
y^{\prime \prime}=y e^{x^{\prime y}}+y \cdot e^{x y}+x y^{2} \cdot e^{x y} \Rightarrow
$$
$$
x=0, y=1 \Rightarrow y^{\prime \prime}=1 \cdot e^{0}+1 \cdot e^{0}+0=2
$$
(3) 求微分方程 $y^{\prime}+\frac{1}{x} y=\frac{1}{x\left(x^{2}+1\right)}$ 的通解 (一般解).
$$
y(x)=\left[\int \frac{1}{x\left(x^{2}+1\right)} e^{\int \frac{1}{x} \mathrm{~ d} x} \mathrm{~ d} x+C\right] e^{-\int \frac{1}{x} \mathrm{~ d} x}=
$$
$$
y(x)=\left[\int \frac{x}{x\left(x^{2}+1\right)} \mathrm{~ d} x+C\right] \cdot \frac{1}{x} \Rightarrow
$$
$$
y(x)=\frac{1}{x} \arctan x+\frac{C}{x}
$$