Let us write the given differential equation in the standard form:

⇒ ……(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 the like variables on one side

⇒

⇒

⇒

We know that:

∫and

Integrating on both sides we get

⇒

⇒

(∵ logC is also an arbitrary constant)

⇒

(∵)

(∵ xloga = loga^x)

Applying exponential on both sides, we get,

⇒

Squaring on both sides we get,

⇒

Since y = vx

we get

⇒

⇒

⇒

Cross multiplying on both sides we get,

⇒ yx² = c²(y + 2x)

∴ The solution to the given differential equation is yx² = c²(y + 2x)