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Help with Differential Eqn

I've been tearing my hair out for the past two hours trying to figure out why I can't do a differential equation.

Here is the problem and the solution:

Attachment 20654

The bottom equation is supposed to be solution, but I have no idea how they got that.

Here is my work trying to solve it (sorry about legibility, had to use a mouse):

Attachment 20655

The (kvo+g) seems to just pop out of nowhere, and I have no idea where that comes from. Can anyone help?

EDIT: to be clear, the Vo is just the constant of integration