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Partial differential equations are great in calculus for making multi-variable equations simpler to solve. Some problems do not have known derivatives or at least in certain levels in your studies, you don't possess the tools needed to find the derivative. So, using partial differential equations, you can break the problem up, and find the partial derivatives and integrals.
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The solution to a differential equation requires integration. With any integration, there is a constant of integration. This constant can only be found by using additional conditions: initial or boundary.
That depends on what type of equation it is because it could be quadratic, simultaneous, linear, straight line or even differential
A differential equation is a tool to certains carrers to find and solve all kinds of problems, in my case i'm a civil engineer and i use this tool to solve problems in the area of hidraulics, and in the area of structures. The differencial ecuations have all kinds of uses in the area of engieneering and in other fields too