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A solution of a differential equation is a relation between the variables (independent and dependent), which is free of derivatives of any order, and which satisfies the differential equation identically. Now let’s get into the details of what ‘differential equations solutions’ actually are!
Differential Equations Solutions
If we consider a general nth order differential equation –

F[x,y,dydx,…..,dnydxn]=0,

where F is a real function of its (n + 2) arguments – x,y,dydx,…..,dnydxn.
Then a function f(x), defined in an interval x ∈ I and having an nth derivative (as well as all of the lower order derivatives) for all x ∈ I; is known as an explicit solution of the given differential equation only if –

F[x,f(x),f′(x),……f(n)(x)]=0, for all x ∈ I.

A relation g(x,y) = 0, is known as the implicit solution of the given differential equation if it defines at least one real function f of the variable x on an interval I such that this function is an explicit solution of the differential equation on this interval, as per the above conditions.

General Solution of a Differential Equation
A General Solution of an nth order differential equation is one that involves n necessary arbitrary constants.

If we solve a first order differential equation by variables separable method, we necessarily have to introduce an arbitrary constant as soon as the integration is performed. Thus you can see that a solution of a differential equation of the first order has 1 necessary arbitrary constant after simplification.

Similarly, the general solution of a second order differential equation will contain 2 necessary arbitrary constants and so on. The general solution geometrically represents an n-parameter family of curves.

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