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I am taking a diff eq course next semester and have been trying to prepare for...

I am taking a diff eq course next semester and have been trying to prepare for it. However, I think i misunderstand something because from the videos and stuff I've seen so far, I don't see a difference between solving a differential equation and finding it's integral. Sorry if this is a stupid question!
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Answer #1

I will hope, This argument resolve your doubts-

Integral will give the most general solution i.e. the family of curve satisfying a given differential equation.

However, a solution of the differential equation may have different to boundary conditions to find out a unique solution. Moreover, in case of the higher-order differential equation(second-order and higher), a particular integral is required apart from the General solution. So integral is not always a solution for the differential equation.

The integral is a general solution of a differential equation. This solution contains the integration constants. The numerical values of these constants depend on the specific problem and are found either from the initial conditions, or from the boundary conditions, or from the initial and boundary conditions. When studying any process for obtaining a specific solution describing this process, always together with the differential equation either initial conditions or boundary conditions (or initial conditions and boundary conditions) are written down.

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