I want to solve the differential equation y &#x2032; </msup> y 2

Bernard Mora

Bernard Mora

Answered question

2022-05-15

I want to solve the differential equation y y 2 = y + x y . I notice that the differential is not in the general form of first order linear equations which is y + P ( x ) y = Q ( x ). In fact the above equation is not linear.
I read that I have to switch the dependent and independent variable of the problem and then the problem will turn into linear first order differential equation. I don't quite understand the concept of switching dependent and independent variable, can someone explain?

Answer & Explanation

Cortez Hughes

Cortez Hughes

Beginner2022-05-16Added 23 answers

I'm not sure, what do you mean by switching variables, but if you simply need to solve it, here's what I'd do
y y 2 = y + x y ( y 3 3 ) = ( x y ) y 3 3 = x y + C
You can leave it as it is in the form of implicit function, or solve for x
x = y 3 3 C 3 y
lifretatox8n

lifretatox8n

Beginner2022-05-17Added 3 answers

To follow on from the comment above. Two things to know
d d x g ( y ) = d g d y y
And
f ( x ) y + f ( x ) y = ( f ( x ) y )
Always look for tricks like the above when solving (or trying to solve) odes.

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