Solve the homogenenous DE :?

Question

dy/dx = (x+3y)/(3x+y).

I have already solved this DE - am not too sure whether I am correct (or not)!

Thx

Answer 1

let y = v x
dy/dx = x dv/dx + v

x dv/dx + v = (x + 3vx) / (3x + vx)
x dv/dx + v = x (1 + 3v) / (x)(3 + v)
x dv/dx + v = (1 + 3v) / (3 + v)
x dv/dx = (1 + 3v) / (3 + v) - v
x dv/dx = [1 + 3v - v (3 + v) ] / (3 + v)
x dv/dx = (1 - v ²) / (3 + v)
∫ (3 + v) / (1 - v ²) dv = ∫ (1/x) dx
(-1/2) log (1 - v ²) = log x + C
(-1/2) log (1 - y ² / x ²) = log x + C
log(1 - y ² / x ²) = - 2 log x + K
log (1 - y ² / x ²) = log x^(-2) + log A
1 - y ² / x ² = A / x ²
y ² / x ² = 1 - A / x ²
y ² = x ² - A

Answer 2

not very pretty but here you go

ln((x+y)/x)-2*ln(-(x-y)/x)= ln(x)+C