T13 欧拉方程

$$D=\frac{\mathrm{d}}{\mathrm{d}t},x=e^t,t=\ln x,\frac{\mathrm{d}t}{\mathrm{d}x}=\frac{1}{x}$$ $$\begin{array}{rl}x{y}^{\prime }& =x\frac{\mathrm{d}y}{\mathrm{d}x}=Dy\\ x^2{y}^{\prime \prime }& =x^2\left(-\frac{1}{x^2}\frac{\mathrm{d}y}{\mathrm{d}t}+\frac{1}{x^2}\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{t}^2}\right)=D^{2}y-Dy\\ x^n{y}^{(n)}& =D(D-1)(D-2)\cdots (D-n+1)y\end{array}$$