SilverKing
New member
- Joined
- Dec 25, 2013
- Messages
- 23
Hi everyone,
I'm facing some troubles with eliminating constants to make the differential equation from this ordinary equation y=ax^2 - bx + c, where a, b and c are constants.
I'm familiar with eliminating two constants at most like the following example:
So, depending on the preceding example, and since we have three arbitrary constants, we differentiate three time:
y=ax^2 - bx + c
y'=2ax -b
y''=2a
y'''=0
So, the Wrosnkian would be:
y x^2 -x 1
y' 2x -1 0
y'' 2 0 0
y''' 0 0 0
Which leads to: 2y'''=0
Is that correct?
I'm facing some troubles with eliminating constants to make the differential equation from this ordinary equation y=ax^2 - bx + c, where a, b and c are constants.
I'm familiar with eliminating two constants at most like the following example:
Determine the differential equation which has the general solution y=c1 e^2x + c2 e^3x, where c1 and c2 are arbitrary constants.
Solution:
Since we have two arbitrary constants, we differentiate y twice:
y'=2 c1 e^2x + 3 c2 e^3x
y''=4 c1 e^4x + 9 c2 e^3x
Using Wronskian determinant to eliminate the constant:
y 1 1
y' 2 3
y'' 4 9
y(18-12)-y'(9-4)+y''(3-2)=0
y''-5y'+6y=0
So, depending on the preceding example, and since we have three arbitrary constants, we differentiate three time:
y=ax^2 - bx + c
y'=2ax -b
y''=2a
y'''=0
So, the Wrosnkian would be:
y x^2 -x 1
y' 2x -1 0
y'' 2 0 0
y''' 0 0 0
Which leads to: 2y'''=0
Is that correct?