In my differential equations book, I have found the following:
Let be the differential equation of first degree 1 and order n (where i are functions of x and y).
Assuming that it is solvable for p, it can be represented as:
equating each factor to Zero, we get n differential equations of first order and first degree.
Let the solution to these n factors be:
Where are arbitrary constants of integration. Since all the c’s can have any one of an infinite number of values, the above solutions will remain general if we replace by a single arbitrary constant c. Then the n solutions (4) can be re-written as
They can be combined to form the general solution as follows:
Now, my question is, whether equation (1) is the most general form of solution to the differential equation.I think the following is the most general form of solution to the differential equation :
If (1) is the general solution, the constant of integration can be found out by only one IVP say, . So, one IVP will give the particular solution. If (2) is the general solution, one IVP might not be able to give the particular solution to the problem.