College math questions and answers

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seiyakou2005n1 2022-05-23 Answered

In my differential equations book, I have found the following:
Let P 0 ( d y d x ) n + P 1 ( d y d x ) n 1 + P 2 ( d y d x ) n 2 + . . . . . . + P n 1 ( d y d x ) + P n = 0 be the differential equation of first degree 1 and order n (where P i i 0 , 1 , 2 , . . . n are functions of x and y).
Assuming that it is solvable for p, it can be represented as:
[ p f 1 ( x , y ) ] [ p f 2 ( x , y ) ] [ p f 3 ( x , y ) ] . . . . . . . . [ p f n ( x , y ) ] = 0
equating each factor to Zero, we get n differential equations of first order and first degree.
[ p f 1 ( x , y ) ] = 0 ,   [ p f 2 ( x , y ) ] = 0 ,   [ p f 3 ( x , y ) ] = 0 ,   . . . . . . . . [ p f n ( x , y ) ] = 0
Let the solution to these n factors be:
F 1 ( x , y , c 1 ) = 0 ,   F 2 ( x , y , c 2 ) = 0 ,   F 3 ( x , y , c 3 ) = 0 ,   . . . . . . . . F n ( x , y , c n ) = 0
Where c 1 , c 2 , c 3 . . . . . c n 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 c 1 , c 2 , c 3 . . . . . c n by a single arbitrary constant c. Then the n solutions (4) can be re-written as
F 1 ( x , y , c ) = 0 ,   F 2 ( x , y , c ) = 0 ,   F 3 ( x , y , c ) = 0 ,   . . . . . . . . F n ( x , y , c ) = 0
They can be combined to form the general solution as follows:
F 1 ( x , y , c )   F 2 ( x , y , c )   F 3 ( x , y , c )   . . . . . . . . F n ( x , y , c ) = 0                         ( 1 )
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 :
F 1 ( x , y , c 1 )   F 2 ( x , y , c 2 )   F 3 ( x , y , c 3 )   . . . . . . . . F n ( x , y , c n ) = 0                         ( 2 )
If (1) is the general solution, the constant of integration can be found out by only one IVP say, y ( 0 ) = 0. 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.

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