An innocuous at first look differential equation...

Question

Solve the equation

( 9 x −4 y +4 ) dx − ( 2 x −y +1 ) dy = 0

Answer 1

x = u + x₀; y = v + y_0;

9x₀ - 4y₀ = - 4; x₀ = 0

-2x₀ + y₀ = 1; y₀ = 1;

x = u, y = v + 1

dx = du, dy = dv

(9u - 4v)du - (2u - v)dv = 0;

du/dv = (2u - v)/(9u - 4v) = 0; (1)

u = vs, where s is new function of v.

du/dv = s + vds/dv. We substitute in (1): s + vds/dv = (2vs - v)/(9vs - 4v);

vds/dv = (2s - 1)/9s - 4) - s; vds/dv = (2s - 1 - 9s² + 4s)/(9s - 4); dv/v = - (9s - 4)/(3s - 1)²ds;∫

∫dv/v = ∫(1/(3s - 1)² - 3/(3s - 1))ds; lnIvI = - 1/[3(3s - 1)] - lnI3s - 1I + C.

Then back to x and y, s = u/v = x/(y - 1), v = y - 1, u = x