Generalization of integrating factor? For example, if we have $y'+p(x)y=q(x)$, we can obtain $\mu(x)=e^{\int p(x)dx}$ as integrating factor. My question is: exist a generalization of this? ie, if I have $y^{(n)}+p_{n-1}(x)y^{(n-1)}+...+p_0(x)y=q(x)$, exist a "integrating factor" for this?
 A: Integrating factor for higher order equations like $y^{(n)}+p_{n-1}(x)y^{(n-1)}+...+p_0(x)y=q(x)$ does exist, for example one integrating factor of $y''+V(y)=0$ is $y'$, because $y'y''+V(y)y'=0$ can be integrated to get 
$$ \frac{1}{2}y'^2 + \int V(y) = C,$$
from which you can write $y'$ in terms of $y$ and integrate again.
In general the integrating factor is a function $\mu\big(x,y,y',\cdots,y^{(n-1)}\big)$ such that there exists another function $\Psi(x,y,y',\cdots,y^{(n-1)})$, such that
$$ \frac{d}{dx}\Psi(x,y,\cdots,y^{(n-1)}) = \mu\big(x,y,\cdots,y^{(n-1)}\big)\big(y^{(n)}+p_{n-1}(x)y^{(n-1)}+...+p_0(x)y-q(x)\big),$$
where $\frac{d}{dx}$ is the total derivative (like $\frac{d}{dx}(x+y+y'^2)=1+y'+2y'y''$) instead of the partial derivative. For higher order equations, you can find integrating factors systematically by using the so called symmetry methods originated by Norwegian mathematician Sophus Lie. Details can be found in books like Symmetry and integration methods for differential equations by Bluman and Anco or Symmetry Methods for Differential Equations by Hydon (the function $\Psi$ is usually called a first integral). 
