My question is the following: How should I factorize a differential equation from the first order

Timiavawsw9

Timiavawsw9

Answered question

2022-05-28

My question is the following:
How should I factorize a differential equation from the first order but with higher degrees?
Imagine I have:
x 2 ( y ) 2 + x y y 6 y 2 = 0
Is it okay if I replace y with p and just solve the equation for p with x and y being constants?
Thanks in advance!

Answer & Explanation

tatoas9f

tatoas9f

Beginner2022-05-29Added 11 answers

Yes, that is a valid method. Doing this via factorization, you can divide your differential equation by x 2 to obtain:
( y ) 2 + y x y 6 y 2 x 2 = 0
If you want to make the factorization easier to visualize, you can let u = y x for example. Doing this should give you:
( y + 3 y x ) ( y 2 y x ) = 0
This will give you two separate first order ODE's to solve:
(1) y = 3 y x y = 2 y x
They are both separable.
Alternatively, you can use the quadratic formula on your original ODE to solve for y :
y = x y ± ( x y ) 2 + 24 x 2 y 2 2 x 2 = y ± 25 y 2 2 x = y ± 5 y 2 x
Giving you the same two first-order ODE's obtained on (1).

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