I'm looking for a good book that serves well as an introduction to the study of differential equations. I don't want it to be about applications since I'm currently not particularly interested in anything but pure mathematics. I want topics like existence, uniqueness, Laplace transformation, solutions in terms of power series and moderately difficult non-linear equations to be covered rigorously. I want many assignments as well. A bonus would be if Fourier series and simple partial differential equations are covered but this is not mandatory. Preferably I would like the price to not exceed \$$100$. If there is a free pdf that would be wonderful.
Not a duplicate since I am more specific with what I want.