Question

solve the following DE by using the appropriate substitution:

(y^2 + yx) dx - x^2 dy = 0

Answer 1

hint:
I can rewrite your equation as below:
\[\Large \frac{{dy}}{{dx}} = \frac{y}{x} + {\left( {\frac{y}{x}} \right)^2}\]

Answer 2

@MarcLeclair

Answer 3

so, please try this substitution:
\[\Large \frac{y}{x} = u\left( x \right)\]
where u(x) is a new function of the variable x

Answer 4

my question was centered more towards how do i know the substitution is possible. like M(tx,ty) = t m(x,y) but in this case the M doesnt have the same degree

Answer 5

you know this is homogeneous from the test you provided

(y^2 + yx) dx - x^2 dy = 0
dy/dx = (y^2 + yx)/x^2 = f(x,y)
f(kx,ky) = (k^2y^2 + kykx)/k^2x^2 = (y^2 + yx)/x^2 = f(x,y)

you can also tell by inspection as the polynomials, ignoring the y' [or dy, dx], are of order 2.

so v = y/x is the sub to use.