I'm wondering how can I integrate this matrix differential equation:
$$ \dot{L_{ij}}(t) = K_{im}L_{mj}(t) \,\, . $$
My gess is to perform a separation of variable and then integrate both sides, like this,
$$ \int \frac{dL_{ij}(t)}{L_{mj}(t)} = \int K_{im} dt \,\, . $$
Is that method right? If yes, How could I perform the integration in the left hand of the equation above? I'm finding difficult in treating with the indices ...
On
The repeated index $m$ just means a summation over all possible values that index can take
$$ K_{im}L_{mj} = K_{i1}L_{ij} + K_{i2}L_{2j} + \cdots $$
So you cannot divide by $K_{lm}$ as you suggest. This is just a coupled system of differential equations. Depending on the nature of $K$ (constant, $t$-dependent, $\dots$) there are ways of solving the system.
Such a matrix ODE $\dot{L}=KL$ is just a system of separate ODEs $\dot{\mathbf{x}}_i = K \mathbf{x}_i$, one for each $i$, where $\mathbf{x}_i$ is the $i$th column of $L$.
And how to solve an equation of the form $\dot{\mathbf{x}} = K \mathbf{x}$ with constant $K$ is covered in almost every ODE textbook (or on Wikipedia).