Given differential equation is:

⇒ (x² – y²)dx – 2xydy = 0

⇒ (x² – y²)dx = 2xydy

⇒ ……(1)

Homogeneous equation: A equation is said to be homogeneous if f(zx,zy) = zⁿf(x,y) (where n is the order of the homogeneous equation).

Let us assume

⇒

⇒

⇒

⇒

⇒ f(zx,zy) = z⁰f(x,y)

So, given differential equation is a homogeneous differential equation.

We need a substitution to solve this type of linear equation and the substitution is y = vx.

Let us substitute this in (1)

⇒

We know that:

⇒

⇒

⇒

⇒

⇒

⇒

Bringing Like variables on same sides we get,

⇒

⇒

⇒

We know that:

Integrating on both sides, we get,

⇒

⇒

(∵ logC is an arbitrary constant)

Multiplying with -3 on both sides we get,

⇒ log|1-3v²| = -3logx + 3logC

⇒

(∵ )

⇒

(∵ alogx = logx^a)

⇒

Applying exponential on both sides we get,

⇒

Since y = vx, we get,

⇒

⇒

⇒

⇒

Cross multiplying on both sides we get,

⇒ x(x² – 3y²) = c³

⇒ x³ – 3xy² = K (say any arbitrary constant)

∴ The solution for the differential equation is x³ – 3xy² = K