Is the first order equation linear in terms of y? in terms of x?

Question

I was looking at an example in my differentials text book and I was confused by how it changes equations in terms of either x or y to see whether its linear.

Linear and Nonlinear ODE's

Given:

Is this equation linear in terms of y

Yes! This equation is linear in the variable y by writing the alternative form

From what I can see all they have done is change the equation from Leibnitz notation to prime notation. Then moved the terms without y to the right side of the equals side in order to make it a linear styled equation.

Whats confusing me is what exactly is happening to the dx after (y-x)? And how would I rewrite this equations to see if it is linear in terms of x?

Im not sure how important it is to understand this, but your help is greatly appreciated.

Thanks

Answer 1

See below the corrected "alternative form". It isn't linear, due to the term $yy'$.

Considering the function $x(y)$ instead of $y(x)$ : $$xx'-yx'-4xy=0 \quad\text{where } x'=\frac{dx}{dy}$$ Also, it isn't linear, due to the term $xx'$.

Note : There is no closed form for the solution of the ODE.