Your understanding of these things is a little wrong As a part of higher mathematics with ordinary differential equations, you need to use the knowledge of differential and integral of functions, especially integral. You must learn indefinite integral well, because indefinite integral is needed to solve any differential equation, which is the foundation.

As for mathematical analysis, there are only talents majoring in mathematics in universities. It's hard. It doesn't need research. As for analytic geometry, as long as you have a high school level and know what tangent and normal mean, it is enough to learn differential equations.