Analytically tractable Ramsey model: how to solve ODE for optimal trajectories
The differential equation 

$$\dot k =  \frac{1}{\sigma} k^\alpha - \delta k$$

has the structure of a Bernoulli equation. We solve it by the following transformation steps:

1) Mulitply throughout by $k^{-\alpha}$:

$$k^{-\alpha}\dot k =  \frac{1}{\sigma}  - \delta k^{1-\alpha} \tag{1}$$

2) Define the variable 
$$z \equiv k^{1-\alpha} \implies \dot z = (1-\alpha)k^{-a}\dot k \tag{2}$$

3) Combine to get

$$(1),(2) \implies \frac {1}{1-\alpha}\dot z = \frac{1}{\sigma} - \delta z $$

$$\implies \dot z + (1-\alpha)\delta z = \frac {1-\alpha}{\sigma}$$

This is a standard first-order differential equation with constant coefficients. Solve it and then reverse the change of variable to get the result.

PS: The above requires that $k \neq 0$ everywhere, which is the economically meaningful case.