More Differential Equations????

Question

7) Determine if the following differential equation is homogeneous

(4y^2 + 4xy + 8x^2)dx + (4x^2 + xy) * tan(4* (y/x))dy = 0

Answer either n (for no, it is not homogeneous), or give the value of the degree of homogeneity.

8) Solve the differential equation

dy/dx = (2x^2 - 4y^2) / (xy)

Answer 1

Very simply put, homogeneous functions are functions where the sum of the powers of every term are the same. So the first function below is homogeneous of degree 3, the second and third are not homogeneous

eg

1) f(x,y) = x^3+4x^2 y+3y^3    Homogenous of order 3

2) f(x,y) = x^3+y                   y term degree is 1 so the equation is not homogenous

3)f(x,y)  X^2+y^2+1              the term  1 has degree 0

In your case

(4y^2 + 4xy + 8x^2)dx + (4x^2 + xy) * tan(4* (y/x))dy = 0  Homogenous degree 2

8)

the equation is homogenous equation with degree 2

put y = vx

results in     dy = dv+dx  ie     dy/dx  = dv/dx+1

dy/dx = (2x^2 - 4y^2) / (xy)

dv/dx+1  = (2x^2 - 4v^2x^2)/vx^2   removing x^2 common

dv/dx+1 = (2-4v^2)/v

dv/dx    = (2-4v^2-v)/v

v/(2-4v^2-v) dv   = dx

integrating both sides

we get the solution