Differential equations questions and answers

Recent questions in Differential Equations
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.

Bacille John Purca 2022-05-18

 

 

Bacille John Purca 2022-05-18

 

 

Bacille John Purca 2022-05-18

 

 

Speaking of differential equations, these are used not only by those students majoring in Physics because solving differential equations is also quite common in Statistics and Financial Studies. Explore the list of questions and examples of equations to get a basic idea of how it is done.

These answers below are meant to provide you with the starting points as you work with your differential equations. If you need specific help or cannot understand the rules behind the answers that are presented below, start with a simple equation and learn with the provided solutions..