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It's not a matter of one being more difficult than the other- Topics from Calculus III are used in Differential equations (partial derivatives, exact differentials, etc.). Calculus III can be taken at the same time, but that is harder. Calculus III should be a prerequisite for Differential Equations.
What To Do With Them? On its own, a Differential Equation is a wonderful way to express something, but is hard to use. So we try to solve them by turning the Differential Equation into a simpler equation without the differential bits, so we can do calculations, make graphs, predict the future, and so on.
Partial differential equations (Pies) have just one small change from ordinary differential equations — but it makes it significantly harder. In general the vast majority cannot be solved analytically. But a small class of special partial differential equations can be solved analytically.
In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.
The first three are 5000-level courses (suitable to be taken as soon as Master's-level courses are completed); but differential geometry is a 6000-level course, typically taken a year after the others. It's not intrinsically harder; it just requires grounding in the other three fields.
A differential equation is an equation composed of differential operators. The solution to a differential equation is a function. So just as in algebra you use arithmetic as a tool to solve for the unknown, in a differential equation you use calculus as a tool to solve for the unknown. This is a differential equation.
The short answer that differential equations (ordinary and partial), not to mention integral equations, are calculus at their root, and require many of the base skills taught in Calculus I, Calculus II, and Calculus III. It should not be surprising the tools of calculus come to bear to solve such problems.
In calculus, the differential represents the principal part of the change in a function y = f(x) with respect to changes in the independent variable. The differential Dy is defined by. Where is the derivative of f with respect to x, and DX is an additional real variable (so that Dy is a function of x and DX).
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