Given Differential equation is:

⇒

⇒ ……(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 side we get,

⇒

⇒

We know that:

⇒

⇒ -log(1-v²) = logx + logC

⇒ log(1-v²)^-1 = log(Cx)

(∵ alogx = logx^a)

(∵ loga + logb = logab)

⇒

Applying exponential on both sides, we get,

⇒

Since y = vx, we get,

⇒

⇒

⇒

⇒

Cross multiplying on both sides we get,

⇒ x = C(x² – y²)

∴ The solution for the given Differential equation is x = C(X²-y²)