不能用公式也不能降阶的微分方程怎么计算？可以尝试进行变量分离——但如果变量分离不了呢？那就先对影响分离的部分作整体代换

一、题目

已知，通过点 $(1,0)$ 的曲线 $y=y(x)$ 上每一点 $(x, y)$ 处切线的斜率等于 $1+\frac{y}{x}+\left(\frac{y}{x}\right)^{2}$, 则此曲线的方程是多少？

难度评级：

二、解析

$$
\left\{\begin{array}{l}y^{\prime}=1+\frac{y}{x}+\left(\frac{y}{x}\right)^{2} \\ y(1)=0\end{array} \Rightarrow u=\frac{y}{x} \Rightarrow\right.
$$

$$
y=u x \Rightarrow \frac{\mathrm{d} y}{\mathrm{d} x}=u+\frac{\mathrm{d} u}{\mathrm{d} x} x \Rightarrow
$$

$$
\left\{\begin{array}{l}u+x \frac{\mathrm{d} u}{\mathrm{d} x}=1+u+u^{2} \\ u(1)=0\end{array}\right.
$$

$$
\left\{\begin{array}{l}x \frac{\mathrm{d} u}{\mathrm{d} x}=1+u^{2} \\ u(1)=0 \end{array}\right. \Rightarrow\left\{\begin{array}{l}\frac{\mathrm{d} u}{\mathrm{d} x}=\frac{1+u^{2}}{x} \\ u(1)=0\end{array}\right.
$$

$$
\left\{\begin{array}{l}\frac{1}{1+u^{2}} \mathrm{d} u=\frac{1}{x} \mathrm{d} x \\ u(1)=0\end{array} \Rightarrow\right.
$$

$$
\left\{\begin{array}{l}\frac{1}{1+u^{2}} \mathrm{d} u=\int \frac{1}{x} \mathrm{d} x \\ u(1)=0\end{array}\right.
$$

$$
\left\{\begin{array}{l}\arctan u=\ln |x|+C \\ u(1)=0\end{array}\right.
$$

$$
0=0+C \Rightarrow C=0 \Rightarrow
$$

$$
\arctan u=\ln |x| \Rightarrow
$$

$$
\arctan \frac{y}{x}=\ln |x| \Rightarrow \frac{y}{x}=\tan (\ln |x|) \Rightarrow
$$

$$
y=x \tan (\ln |x|)
$$

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