Please solve the homogeneous ode.

Question

(x^2 -y^2)dx - 3xy dy = 0?

Answer 1

(x² -y²)dx - 3xy dy = 0

Let y = vx

dy = x dv + v dx

(x² -(vx)²)dx - 3x(vx)(x dv + v dx)= 0

(x² - (vx)²)dx - 3x²v (x dv + v dx)= 0

x²dx - v²x²dx - 3x³vdv - 3x²v² dx= 0

(x² - v²x² - 3x²v² )dx - 3x³vdv= 0

(x² - 4x²v² )dx = 3x³vdv

x²(1 - 4v² )dx = 3x³vdv

(x²/x³)dx = [3v/(1 - 4v² )]dv

(1/x)dx = [3v/(1 - 4v² )]dv

Apply integration both sides.

∫(1/x)dx = 3∫[v/(1 - 4v² )]dv

∫ (1/x)dx = (-3/8)∫[-8v/(1 - 4v² )]dv

ln| x | = –3/8 ln(1 - 4v²) + lnC₁

8ln| x⁸ | = ln[1/(1 - 4v²)³] +8lnC₁

8ln| x⁸ | = ln[1/(1 - 4v²)³] +lnc

ln| x⁸ | = ln[c/(1 - 4v²)³]

x⁸  = c/(1 - 4v²)³

Substitute : y = v/x

x⁸  = c/(1 - 4(y/x)²)³

x⁸  = cx²/(x² - 4y²)³

x⁶  = c/(x² - 4y²)³

x⁶(x² - 4y²)³  = c

Where c is constant.

Solution : x⁶(x² - 4y²)³ = c.