I am planning to take Differential equations next semester, but due to a timetable issue I want to study most of it this summer in my spare time to make it easier.
These are the Topics that will be included, which I think represent about half of the Differential equations in other universities:
Ordinary differential equations. Explicitly solvable equations, exact and linear equations. Well-posedness of the initial value problem, existence, uniqueness, continuous dependence on initial values. Approximate solution methods. Linear systems of equations, variational system. Elements of stability theory, stability, asymptotic stability, Lyapunov functions, stability by the linear approximation. Phase portraits of planar autonomous equations. Laplace transform, application to solve differential equations. Discrete-time dynamical systems.
I am not only looking for textbooks with rigorous exercise sets, any books are welcome, heavy in theory ones as well.