Solve the DE using homogeneous equation

Question

(x + 2y) dx - 3y dy = 0

a. x^4 - 6 x^2y^2 + 8 xy^3 - 3 y^4 = C

b. 3 x^4 - 8 x^2y^2 + 6 xy^3 = C

c. x^4 + 3x^2y^2 - 4 xy^3 + 3 y^4 = C

d. x^4 + 3x^2y^2 - 6xy^3 = C

Answer 1

dy/dx = (x + 2y)/(3y); y = vx; dy/dx = v + xdv/dx; v + xdv/dx = (x + 2vx)/(3vx); xdv/dx = (1 + 2v)/(3v) - v;

xdv/dx = (1 + 2v - 3v²)/(3v; 3vdv/(1 + 2v - 3v²) = dx/x; ∫3vdv/(1 - v)(3v + 1) = ∫dx/x; ∫3/4[1/(1 - v) - 1/(3v +1)] =

∫dx/x; 3/4[- ln(v - 1) - 1/3ln(3v + 1) = lnx - 3/4lnC; (v - 1)^-3/4 (3v +1)^-1/4 = xC^-3/4; (y/x - 1)^-3/4(3y/x + 1)^-1/4 =

(y/x - 1)(3y/x + 1)^1/3 = C/x^4/3; (y/x - 1)³(3y/x + 1) = C/x⁴; x⁴(y³/x³- 3y²/x² + 3y/x - 1)(3y/x + 1) = C;

3y⁴- 9y³x + 9y²x²- 3yx³ + y³x - 3y²x² + 3yx³ - x⁴ = C; x⁴ + 8y³x - 6y²x² - 3y⁴ = C

Answer is b.