# 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

## 二、解析

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

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

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

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

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

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

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

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

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

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

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