欧拉方程（二阶）

$x^n{y}^{(n)}+a_1{x}^{n-1}{y}^{(n-1)}+\cdot \cdot \cdot +a_{n-1}x{y}^{\prime }+a_{n}y=f(x)$

形如上式的方程称为 欧拉方程

二阶非齐次方程

$x^2{y}^{\prime \prime }(x)+a_{1}x{y}^{\prime }(x)+a_{2}y(x)=f(x)$

求通解

若 $x>0$，令 $x=e^t$，则

\begin{gather} y = y(x) = y(e^t) = y(t) = y\\ \\

y’ = \frac{dy}{dx} = \frac{dy}{dt} \frac{dt}{dx} = \frac{dy}{dt} \frac{1}{\frac{dx}{dt}} = \frac{dy}{dt} \frac{1}{e^t}\ \
y” = \frac{d^2 y}{dx^2} = \frac{d}{dx} \left( \frac{dy}{dx} \right) = \frac{d}{dx} \left( \frac{dy}{dt} \frac{1}{e^t} \right) = \frac{d}{dt} \left( \frac{dy}{dt} \frac{1}{e^t} \right) \frac{dt}{dx} =\frac{d^2y}{dt^2} \frac{1}{e^{2t}}\ \

\end{gather}

则有 $I=y$