Is there a general method for solving a linear homogeneous ordinary differential equation? I am wondering if there is a general method for finding the solutions of a linear ODE of the form :
$Ly=0$
Note : the coefficients of the homogenous LODE can be variable
 A: If you mean equations of the form:
$$
y^{(n)}(t) - a_{n-1}(t)y^{(n-1)}(t)-\cdots-a_0(t)y(t) = 0
$$
with variable coefficients $a_{0}(t),\dots,a_{n-1}(t)$, then one approach could be the following. You can write the ODE in matrix form as
$$
\frac{d\mathbf{y}}{dt} = A(t)\mathbf{y}
$$
with $\mathbf{y} = [y,y^{(1)},\dots,y^{(n-1)}]$ and
$$
A(t) = \begin{bmatrix}
0 & 1 & 0 &\cdots &0 \\
\vdots & \ddots & \ddots & \ddots & \vdots \\
0&\cdots&0 & \ddots & 0 \\
0 & \cdots & 0 & 0 & 1 \\
a_0(t) &\cdots & \cdots & \cdots & a_{n-1}(t)\\
\end{bmatrix}
$$
Thus, the solution to the ODE is given by $$\mathbf{y}(t) = \Phi(t,t_0)\mathbf{y}(t_0)$$
for some initial condition $\mathbf{y}(t_0)$, where $\Phi(t,t_0)$ is called the state transition matrix, and can be computed (or approximated if you want) from $A(t)$ only (under suitable conditions on the coeficients) using the Peano-Baker series here. If $A(t) = A$ is constant, the state-transition matrix reduces to the well know exponential matrix $\Phi(t,t_0) = \exp(A(t-t_0))$.
